Canonical uniformity

The diagonal of ${\displaystyle X}$ is the set^[1]

$${\displaystyle \Delta _{X}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{(x,x):x\in X\}}$$

and for any ${\displaystyle N\subseteq X,}$ the canonical entourage/vicinity around ${\displaystyle N}$ is the set

$${\displaystyle {\begin{alignedat}{4}\Delta _{X}(N)~&~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{(x,y)\in X\times X~:~x-y\in N\}=\bigcup _{y\in X}[(y+N)\times \{y\}]\\&=\Delta _{X}+(N\times \{0\})\end{alignedat}}}$$

where if ${\displaystyle 0\in N}$ then ${\displaystyle \Delta _{X}(N)}$ contains the diagonal ${\displaystyle \Delta _{X}(\{0\})=\Delta _{X}.}$

If ${\displaystyle N}$ is a symmetric set (that is, if ${\displaystyle -N=N}$), then ${\displaystyle \Delta _{X}(N)}$ is symmetric, which by definition means that ${\displaystyle \Delta _{X}(N)=\left(\Delta _{X}(N)\right)^{\operatorname {op} }}$ holds where ${\displaystyle \left(\Delta _{X}(N)\right)^{\operatorname {op} }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\left\{(y,x):(x,y)\in \Delta _{X}(N)\right\},}$ and in addition, this symmetric set's composition with itself is:

$${\displaystyle {\begin{alignedat}{4}\Delta _{X}(N)\circ \Delta _{X}(N)~&~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\left\{(x,z)\in X\times X~:~{\text{ there exists }}y\in X{\text{ such that }}x,z\in y+N\right\}=\bigcup _{y\in X}[(y+N)\times (y+N)]\\&=\Delta _{X}+(N\times N).\end{alignedat}}}$$

If ${\displaystyle {\mathcal {L}}}$ is any neighborhood basis at the origin in ${\displaystyle (X,\tau )}$ then the family of subsets of ${\displaystyle X\times X:}$

$${\displaystyle {\mathcal {B}}_{\mathcal {L}}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\left\{\Delta _{X}(N):N\in {\mathcal {L}}\right\}}$$

is a prefilter on ${\displaystyle X\times X.}$ If ${\displaystyle {\mathcal {N}}_{\tau }(0)}$ is the neighborhood filter at the origin in ${\displaystyle (X,\tau )}$ then ${\displaystyle {\mathcal {B}}_{{\mathcal {N}}_{\tau }(0)}}$ forms a base of entourages for a uniform structure on ${\displaystyle X}$ that is considered canonical.^[2] Explicitly, by definition, the canonical uniformity on ${\displaystyle X}$ induced by ${\displaystyle (X,\tau )}$^[2] is the filter ${\displaystyle {\mathcal {U}}_{\tau }}$ on ${\displaystyle X\times X}$ generated by the above prefilter:

$${\displaystyle {\mathcal {U}}_{\tau }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~{\mathcal {B}}_{{\mathcal {N}}_{\tau }(0)}^{\uparrow }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\left\{S\subseteq X\times X~:~N\in {\mathcal {N}}_{\tau }(0){\text{ and }}\Delta _{X}(N)\subseteq S\right\}}$$

where ${\displaystyle {\mathcal {B}}_{{\mathcal {N}}_{\tau }(0)}^{\uparrow }}$ denotes the upward closure of ${\displaystyle {\mathcal {B}}_{{\mathcal {N}}_{\tau }(0)}}$ in ${\displaystyle X\times X.}$ The same canonical uniformity would result by using a neighborhood basis of the origin rather the filter of all neighborhoods of the origin. If ${\displaystyle {\mathcal {L}}}$ is any neighborhood basis at the origin in ${\displaystyle (X,\tau )}$ then the filter on ${\displaystyle X\times X}$ generated by the prefilter ${\displaystyle {\mathcal {B}}_{\mathcal {L}}}$ is equal to the canonical uniformity ${\displaystyle {\mathcal {U}}_{\tau }}$ induced by ${\displaystyle (X,\tau ).}$

Cauchy net

The general theory of uniform spaces has its own definition of a "Cauchy prefilter" and "Cauchy net". For the canonical uniformity on ${\displaystyle X,}$ these definitions reduce down to those given below.

Suppose ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}$ is a net in ${\displaystyle X}$ and ${\displaystyle y_{\bullet }=\left(y_{j}\right)_{j\in J}}$ is a net in ${\displaystyle Y.}$ The product ${\displaystyle I\times J}$ becomes a directed set by declaring ${\displaystyle (i,j)\leq \left(i_{2},j_{2}\right)}$ if and only if ${\displaystyle i\leq i_{2}}$ and ${\displaystyle j\leq j_{2}.}$ Then

$${\displaystyle x_{\bullet }\times y_{\bullet }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\left(x_{i},y_{j}\right)_{(i,j)\in I\times J}}$$

denotes the (Cartesian) product net, where in particular ${\textstyle x_{\bullet }\times x_{\bullet }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\left(x_{i},x_{j}\right)_{(i,j)\in I\times I}.}$ If ${\displaystyle X=Y}$ then the image of this net under the vector addition map ${\displaystyle X\times X\to X}$ denotes the sum of these two nets:^[3]

$${\displaystyle x_{\bullet }+y_{\bullet }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\left(x_{i}+y_{j}\right)_{(i,j)\in I\times J}}$$

and similarly their difference is defined to be the image of the product net under the vector subtraction map ${\displaystyle (x,y)\mapsto x-y}$:

$${\displaystyle x_{\bullet }-y_{\bullet }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\left(x_{i}-y_{j}\right)_{(i,j)\in I\times J}.}$$

In particular, the notation ${\displaystyle x_{\bullet }-x_{\bullet }=\left(x_{i}\right)_{i\in I}-\left(x_{i}\right)_{i\in I}}$ denotes the ${\displaystyle I^{2}}$-indexed net ${\displaystyle \left(x_{i}-x_{j}\right)_{(i,j)\in I\times I}}$ and not the ${\displaystyle I}$-indexed net ${\displaystyle \left(x_{i}-x_{i}\right)_{i\in I}=(0)_{i\in I}}$ since using the latter as the definition would make the notation useless.

A net ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}$ in a TVS ${\displaystyle X}$ is called a Cauchy net^[4] if

$${\displaystyle x_{\bullet }-x_{\bullet }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\left(x_{i}-x_{j}\right)_{(i,j)\in I\times I}\to 0\quad {\text{ in }}X.}$$

Explicitly, this means that for every neighborhood ${\displaystyle N}$ of ${\displaystyle 0}$ in ${\displaystyle X,}$ there exists some index ${\displaystyle i_{0}\in I}$ such that ${\displaystyle x_{i}-x_{j}\in N}$ for all indices ${\displaystyle i,j\in I}$ that satisfy ${\displaystyle i\geq i_{0}}$ and ${\displaystyle j\geq i_{0}.}$ It suffices to check any of these defining conditions for any given neighborhood basis of ${\displaystyle 0}$ in ${\displaystyle X.}$ A Cauchy sequence is a sequence that is also a Cauchy net.

If ${\displaystyle x_{\bullet }\to x}$ then ${\displaystyle x_{\bullet }\times x_{\bullet }\to (x,x)}$ in ${\displaystyle X\times X}$ and so the continuity of the vector subtraction map ${\displaystyle S:X\times X\to X,}$ which is defined by ${\displaystyle S(x,y)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~x-y,}$ guarantees that ${\displaystyle S\left(x_{\bullet }\times x_{\bullet }\right)\to S(x,x)}$ in ${\displaystyle X,}$ where ${\displaystyle S\left(x_{\bullet }\times x_{\bullet }\right)=\left(x_{i}-x_{j}\right)_{(i,j)\in I\times I}=x_{\bullet }-x_{\bullet }}$ and ${\displaystyle S(x,x)=x-x=0.}$ This proves that every convergent net is a Cauchy net. By definition, a space is called complete if the converse is also always true. That is, ${\displaystyle X}$ is complete if and only if the following holds:

whenever ${\displaystyle x_{\bullet }}$ is a net in ${\displaystyle X,}$ then ${\displaystyle x_{\bullet }}$ converges (to some point) in ${\displaystyle X}$ if and only if ${\displaystyle x_{\bullet }-x_{\bullet }\to 0}$ in ${\displaystyle X.}$

A similar characterization of completeness holds if filters and prefilters are used instead of nets.

A series ${\displaystyle \sum _{i=1}^{\infty }x_{i}}$ is called a Cauchy series (respectively, a convergent series) if the sequence of partial sums ${\displaystyle \left(\sum _{i=1}^{n}x_{i}\right)_{n=1}^{\infty }}$ is a Cauchy sequence (respectively, a convergent sequence).^[5] Every convergent series is necessarily a Cauchy series. In a complete TVS, every Cauchy series is necessarily a convergent series.

Cauchy filter and Cauchy prefilter

A prefilter ${\displaystyle {\mathcal {B}}}$ on a topological vector space ${\displaystyle X}$ is called a Cauchy prefilter^[6] if it satisfies any of the following equivalent conditions:

1. ${\displaystyle {\mathcal {B}}-{\mathcal {B}}\to 0}$ in ${\displaystyle X.}$
   • The family ${\displaystyle {\mathcal {B}}-{\mathcal {B}}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{B-C:B,C\in {\mathcal {B}}\}}$ is a prefilter.
   • Explicitly, ${\displaystyle {\mathcal {B}}-{\mathcal {B}}\to 0}$ means that for every neighborhood ${\displaystyle N}$ of the origin in ${\displaystyle X,}$ there exist ${\displaystyle B,C\in {\mathcal {B}}}$ such that ${\displaystyle B-C\subseteq N.}$
2. ${\displaystyle \{B-B:B\in {\mathcal {B}}\}\to 0}$ in ${\displaystyle X.}$
   • The family ${\displaystyle \{B-B:B\in {\mathcal {B}}\}}$ is a prefilter equivalent to ${\displaystyle {\mathcal {B}}-{\mathcal {B}}}$ (equivalence means these prefilters generate the same filter on ${\displaystyle X}$).
   • Explicitly, ${\displaystyle \{B-B:B\in {\mathcal {B}}\}\to 0}$ means that for every neighborhood ${\displaystyle N}$ of the origin in ${\displaystyle X,}$ there exists some ${\displaystyle B\in {\mathcal {B}}}$ such that ${\displaystyle B-B\subseteq N.}$
3. For every neighborhood ${\displaystyle N}$ of the origin in ${\displaystyle X,}$ ${\displaystyle {\mathcal {B}}}$ contains some ${\displaystyle N}$-small set (that is, there exists some ${\displaystyle B\in {\mathcal {B}}}$ such that ${\displaystyle B-B\subseteq N}$).^[6]
   • A subset ${\displaystyle B\subseteq X}$ is called ${\displaystyle N}$-small or small of order ${\displaystyle N}$^[6] if ${\displaystyle B-B\subseteq N.}$
4. For every neighborhood ${\displaystyle N}$ of the origin in ${\displaystyle X,}$ there exists some ${\displaystyle x\in X}$ and some ${\displaystyle B\in {\mathcal {B}}}$ such that ${\displaystyle B\subseteq x+N.}$^[6]
   • This statement remains true if "${\displaystyle B\subseteq x+N}$" is replaced with "${\displaystyle x+B\subseteq N.}$"
5. Every neighborhood of the origin in ${\displaystyle X}$ contains some subset of the form ${\displaystyle x+B}$ where ${\displaystyle x\in X}$ and ${\displaystyle B\in {\mathcal {B}}.}$

It suffices to check any of the above conditions for any given neighborhood basis of ${\displaystyle 0}$ in ${\displaystyle X.}$ A Cauchy filter is a Cauchy prefilter that is also a filter on ${\displaystyle X.}$

If ${\displaystyle {\mathcal {B}}}$ is a prefilter on a topological vector space ${\displaystyle X}$ and if ${\displaystyle x\in X,}$ then ${\displaystyle {\mathcal {B}}\to x}$ in ${\displaystyle X}$ if and only if ${\displaystyle x\in \operatorname {cl} {\mathcal {B}}}$ and ${\displaystyle {\mathcal {B}}}$ is Cauchy.^[3]

Complete subset

For any ${\displaystyle S\subseteq X,}$ a prefilter ${\displaystyle {\mathcal {C}}}$ on ${\displaystyle S}$ is necessarily a subset of ${\displaystyle \wp (S)}$; that is, ${\displaystyle {\mathcal {C}}\subseteq \wp (S).}$

A subset ${\displaystyle S}$ of a TVS ${\displaystyle (X,\tau )}$ is called a complete subset if it satisfies any of the following equivalent conditions:

1. Every Cauchy prefilter ${\displaystyle {\mathcal {C}}\subseteq \wp (S)}$ on ${\displaystyle S}$ converges to at least one point of ${\displaystyle S.}$
   • If ${\displaystyle X}$ is Hausdorff then every prefilter on ${\displaystyle S}$ will converge to at most one point of ${\displaystyle X.}$ But if ${\displaystyle X}$ is not Hausdorff then a prefilter may converge to multiple points in ${\displaystyle X.}$ The same is true for nets.
2. Every Cauchy net in ${\displaystyle S}$ converges to at least one point of ${\displaystyle S.}$
3. ${\displaystyle S}$ is a complete uniform space (under the point-set topology definition of "complete uniform space") when ${\displaystyle S}$ is endowed with the uniformity induced on it by the canonical uniformity of ${\displaystyle X.}$

The subset ${\displaystyle S}$ is called a sequentially complete subset if every Cauchy sequence in ${\displaystyle S}$ (or equivalently, every elementary Cauchy filter/prefilter on ${\displaystyle S}$) converges to at least one point of ${\displaystyle S.}$

Importantly, convergence to points outside of ${\displaystyle S}$ does not prevent a set from being complete: If ${\displaystyle X}$ is not Hausdorff and if every Cauchy prefilter on ${\displaystyle S}$ converges to some point of ${\displaystyle S,}$ then ${\displaystyle S}$ will be complete even if some or all Cauchy prefilters on ${\displaystyle S}$ also converge to points(s) in ${\displaystyle X\setminus S.}$ In short, there is no requirement that these Cauchy prefilters on ${\displaystyle S}$ converge only to points in ${\displaystyle S.}$ The same can be said of the convergence of Cauchy nets in ${\displaystyle S.}$

As a consequence, if a TVS ${\displaystyle X}$ is not Hausdorff then every subset of the closure of ${\displaystyle \{0\}}$ in ${\displaystyle X}$ is complete because it is compact and every compact set is necessarily complete. In particular, if ${\displaystyle \varnothing \neq S\subseteq \operatorname {cl} _{X}\{0\}}$ is a proper subset, such as ${\displaystyle S=\{0\}}$ for example, then ${\displaystyle S}$ would be complete even though every Cauchy net in ${\displaystyle S}$ (and also every Cauchy prefilter on ${\displaystyle S}$) converges to every point in ${\displaystyle \operatorname {cl} _{X}\{0\},}$ including those points in ${\displaystyle \operatorname {cl} _{X}\{0\}}$ that do not belong to ${\displaystyle S.}$ This example also shows that complete subsets (and indeed, even compact subsets) of a non-Hausdorff TVS may fail to be closed. For example, if ${\displaystyle \varnothing \neq S\subseteq \operatorname {cl} _{X}\{0\}}$ then ${\displaystyle S=\operatorname {cl} _{X}\{0\}}$ if and only if ${\displaystyle S}$ is closed in ${\displaystyle X.}$

Complete topological vector space

A topological vector space ${\displaystyle X}$ is called a complete topological vector space if any of the following equivalent conditions are satisfied:

1. ${\displaystyle X}$ is a complete uniform space when it is endowed with its canonical uniformity.
   • In the general theory of uniform spaces, a uniform space is called a complete uniform space if each Cauchy filter on ${\displaystyle X}$ converges to some point of ${\displaystyle X}$ in the topology induced by the uniformity. When ${\displaystyle X}$ is a TVS, the topology induced by the canonical uniformity is equal to ${\displaystyle X}$'s given topology (so convergence in this induced topology is just the usual convergence in ${\displaystyle X}$).
2. ${\displaystyle X}$ is a complete subset of itself.
3. There exists a neighborhood of the origin in ${\displaystyle X}$ that is also a complete subset of ${\displaystyle X.}$^[6]
   • This implies that every locally compact TVS is complete (even if the TVS is not Hausdorff).
4. Every Cauchy prefilter ${\displaystyle {\mathcal {C}}\subseteq \wp (X)}$ on ${\displaystyle X}$ converges in ${\displaystyle X}$ to at least one point of ${\displaystyle X.}$
   • If ${\displaystyle X}$ is Hausdorff then every prefilter on ${\displaystyle X}$ will converge to at most one point of ${\displaystyle X.}$ But if ${\displaystyle X}$ is not Hausdorff then a prefilter may converge to multiple points in ${\displaystyle X.}$ The same is true for nets.
5. Every Cauchy filter on ${\displaystyle X}$ converges in ${\displaystyle X}$ to at least one point of ${\displaystyle X.}$
6. Every Cauchy net in ${\displaystyle X}$ converges in ${\displaystyle X}$ to at least one point of ${\displaystyle X.}$

where if in addition ${\displaystyle X}$ is pseudometrizable or metrizable (for example, a normed space) then this list can be extended to include:

7. ${\displaystyle X}$ is sequentially complete.

A topological vector space ${\displaystyle X}$ is sequentially complete if any of the following equivalent conditions are satisfied:

1. ${\displaystyle X}$ is a sequentially complete subset of itself.
2. Every Cauchy sequence in ${\displaystyle X}$ converges in ${\displaystyle X}$ to at least one point of ${\displaystyle X.}$
3. Every elementary Cauchy prefilter on ${\displaystyle X}$ converges in ${\displaystyle X}$ to at least one point of ${\displaystyle X.}$
4. Every elementary Cauchy filter on ${\displaystyle X}$ converges in ${\displaystyle X}$ to at least one point of ${\displaystyle X.}$

Uniform spaces and translation-invariant uniformities

For any subsets ${\displaystyle \Phi ,\Psi \subseteq X\times X,}$ let^[1]

$${\displaystyle \Phi ^{\operatorname {op} }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{(y,x)~:~(x,y)\in \Phi \}}$$

and let

$${\displaystyle {\begin{alignedat}{4}\Phi \circ \Psi ~&~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\left\{(x,z):{\text{ there exists }}y\in X{\text{ such that }}(x,y)\in \Psi {\text{ and }}(y,z)\in \Phi \right\}\\&=~\bigcup _{y\in X}\{(x,z)~:~(x,y)\in \Psi {\text{ and }}(y,z)\in \Phi \}\end{alignedat}}}$$

A non-empty family ${\displaystyle {\mathcal {B}}\subseteq \wp (X\times X)}$ is called a base of entourages or a fundamental system of entourages if ${\displaystyle {\mathcal {B}}}$ is a prefilter on ${\displaystyle X\times X}$ satisfying all of the following conditions:

1. Every set in ${\displaystyle {\mathcal {B}}}$ contains the diagonal of ${\displaystyle X}$ as a subset; that is, ${\displaystyle \Delta _{X}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{(x,x):x\in X\}\subseteq \Phi }$ for every ${\displaystyle \Phi \in {\mathcal {B}}.}$ Said differently, the prefilter ${\displaystyle {\mathcal {B}}}$ is fixed on ${\displaystyle \Delta _{X}.}$
2. For every ${\displaystyle \Omega \in {\mathcal {B}}}$ there exists some ${\displaystyle \Phi \in {\mathcal {B}}}$ such that ${\displaystyle \Phi \circ \Phi \subseteq \Omega .}$
3. For every ${\displaystyle \Omega \in {\mathcal {B}}}$ there exists some ${\displaystyle \Phi \in {\mathcal {B}}}$ such that ${\displaystyle \Phi \subseteq \Omega ^{\operatorname {op} }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{(y,x):(x,y)\in \Omega \}.}$

A uniformity or uniform structure on ${\displaystyle X}$ is a filter ${\displaystyle {\mathcal {U}}}$ on ${\displaystyle X\times X}$ that is generated by some base of entourages ${\displaystyle {\mathcal {B}},}$ in which case we say that ${\displaystyle {\mathcal {B}}}$ is a base of entourages for ${\displaystyle {\mathcal {U}}.}$

For a commutative additive group ${\displaystyle X,}$ a translation-invariant fundamental system of entourages^[7] is a fundamental system of entourages ${\displaystyle {\mathcal {B}}}$ such that for every ${\displaystyle \Phi \in {\mathcal {B}},}$ ${\displaystyle (x,y)\in \Phi }$ if and only if ${\displaystyle (x+z,y+z)\in \Phi }$ for all ${\displaystyle x,y,z\in X.}$ A uniformity ${\displaystyle {\mathcal {B}}}$ is called a translation-invariant uniformity^[7] if it has a base of entourages that is translation-invariant. The canonical uniformity on any TVS is translation-invariant.^[7]

The binary operator ${\displaystyle \;\circ \;}$ satisfies all of the following:

• ${\displaystyle (\Phi \circ \Psi )^{\operatorname {op} }=\Psi ^{\operatorname {op} }\circ \Phi ^{\operatorname {op} }.}$
• If ${\displaystyle \Phi \subseteq \Phi _{2}}$ and ${\displaystyle \Psi \subseteq \Psi _{2}}$ then ${\displaystyle \Phi \circ \Psi \subseteq \Phi _{2}\circ \Psi _{2}.}$
• Associativity: ${\displaystyle \Phi \circ (\Psi \circ \Omega )=(\Phi \circ \Psi )\circ \Omega .}$
• Identity: ${\displaystyle \Phi \circ \Delta _{X}=\Phi =\Delta _{X}\circ \Phi .}$
• Zero: ${\displaystyle \Phi \circ \varnothing =\varnothing =\varnothing \circ \Phi }$

Symmetric entourages

Call a subset ${\displaystyle \Phi \subseteq X\times X}$ symmetric if ${\displaystyle \Phi =\Phi ^{\operatorname {op} },}$ which is equivalent to ${\displaystyle \Phi ^{\operatorname {op} }\subseteq \Phi .}$ This equivalence follows from the identity ${\displaystyle \left(\Phi ^{\operatorname {op} }\right)^{\operatorname {op} }=\Phi }$ and the fact that if ${\displaystyle \Psi \subseteq X\times X,}$ then ${\displaystyle \Phi \subseteq \Psi }$ if and only if ${\displaystyle \Phi ^{\operatorname {op} }\subseteq \Psi ^{\operatorname {op} }.}$ For example, the set ${\displaystyle \Phi ^{\operatorname {op} }\cap \Phi }$ is always symmetric for every ${\displaystyle \Phi \subseteq X\times X.}$ And because ${\displaystyle (\Phi \cap \Psi )^{\operatorname {op} }=\Phi ^{\operatorname {op} }\cap \Psi ^{\operatorname {op} },}$ if ${\displaystyle \Phi }$ and ${\displaystyle \Psi }$ are symmetric then so is ${\displaystyle \Phi \cap \Psi .}$

Topology generated by a uniformity

Relatives

Let ${\displaystyle \Phi \subseteq X\times X}$ be arbitrary and let ${\displaystyle \operatorname {Pr} _{1},\operatorname {Pr} _{2}:X\times X\to X}$ be the canonical projections onto the first and second coordinates, respectively.

For any ${\displaystyle S\subseteq X,}$ define

$${\displaystyle S\cdot \Phi ~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{y\in X:\Phi \cap (S\times \{x\})\neq \varnothing \}~=~\operatorname {Pr} _{2}(\Phi \cap (S\times X))}$$

$${\displaystyle \Phi \cdot S~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{x\in X:\Phi \cap (\{x\}\times S)\neq \varnothing \}~=~\operatorname {Pr} _{1}(\Phi \cap (X\times S))=S\cdot \left(\Phi ^{\operatorname {op} }\right)}$$

where ${\displaystyle \Phi \cdot S}$ (respectively, ${\displaystyle S\cdot \Phi }$) is called the set of left (respectively, right) ${\displaystyle \Phi }$-relatives of (points in) ${\displaystyle S.}$ Denote the special case where ${\displaystyle S=\{p\}}$ is a singleton set for some ${\displaystyle p\in X}$ by:

$${\displaystyle p\cdot \Phi ~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{p\}\cdot \Phi ~=~\{y\in X:(p,y)\in \Phi \}}$$

$${\displaystyle \Phi \cdot p~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\Phi \cdot \{p\}~=~\{x\in X:(x,p)\in \Phi \}~=~p\cdot \left(\Phi ^{\operatorname {op} }\right)}$$

If ${\displaystyle \Phi ,\Psi \subseteq X\times X}$ then ${\textstyle (\Phi \circ \Psi )\cdot S=\Phi \cdot (\Psi \cdot S).}$ Moreover, ${\displaystyle \,\cdot \,}$ right distributes over both unions and intersections, meaning that if ${\displaystyle R,S\subseteq X}$ then ${\displaystyle (R\cup S)\cdot \Phi ~=~(R\cdot \Phi )\cup (S\cdot \Phi )}$ and ${\displaystyle (R\cap S)\cdot \Phi ~\subseteq ~(R\cdot \Phi )\cap (S\cdot \Phi ).}$

Neighborhoods and open sets

Two points ${\displaystyle x}$ and ${\displaystyle y}$ are ${\displaystyle \Phi }$-close if ${\displaystyle (x,y)\in \Phi }$ and a subset ${\displaystyle S\subseteq X}$ is called ${\displaystyle \Phi }$-small if ${\displaystyle S\times S\subseteq \Phi .}$

Let ${\displaystyle {\mathcal {B}}\subseteq \wp (X\times X)}$ be a base of entourages on ${\displaystyle X.}$ The neighborhood prefilter at a point ${\displaystyle p\in X}$ and, respectively, on a subset ${\displaystyle S\subseteq X}$ are the families of sets:

$${\displaystyle {\mathcal {B}}\cdot p~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~{\mathcal {B}}\cdot \{p\}=\{\Phi \cdot p:\Phi \in {\mathcal {B}}\}\qquad {\text{ and }}\qquad {\mathcal {B}}\cdot S~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{\Phi \cdot S:\Phi \in {\mathcal {B}}\}}$$

and the filters on ${\displaystyle X}$ that each generates is known as the neighborhood filter of ${\displaystyle p}$ (respectively, of ${\displaystyle S}$). Assign to every ${\displaystyle x\in X}$ the neighborhood prefilter

$${\displaystyle {\mathcal {B}}\cdot x~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{\Phi \cdot x:\Phi \in {\mathcal {B}}\}}$$

and use the neighborhood definition of "open set" to obtain a topology on ${\displaystyle X}$ called the topology induced by ${\displaystyle {\mathcal {B}}}$ or the induced topology. Explicitly, a subset ${\displaystyle U\subseteq X}$ is open in this topology if and only if for every ${\displaystyle u\in U}$ there exists some ${\displaystyle N\in {\mathcal {B}}\cdot u}$ such that ${\displaystyle N\subseteq U;}$ that is, ${\displaystyle U}$ is open if and only if for every ${\displaystyle u\in U}$ there exists some ${\displaystyle \Phi \in {\mathcal {B}}}$ such that ${\displaystyle \Phi \cdot u~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{x\in X:(x,u)\in \Phi \}\subseteq U.}$

The closure of a subset ${\displaystyle S\subseteq X}$ in this topology is:

$${\displaystyle \operatorname {cl} _{X}S=\bigcap _{\Phi \in {\mathcal {B}}}(\Phi \cdot S)=\bigcap _{\Phi \in {\mathcal {B}}}(S\cdot \Phi ).}$$

Cauchy prefilters and complete uniformities

A prefilter ${\displaystyle {\mathcal {F}}\subseteq \wp (X)}$ on a uniform space ${\displaystyle X}$ with uniformity ${\displaystyle {\mathcal {U}}}$ is called a Cauchy prefilter if for every entourage ${\displaystyle N\in {\mathcal {U}},}$ there exists some ${\displaystyle F\in {\mathcal {F}}}$ such that ${\displaystyle F\times F\subseteq N.}$

A uniform space ${\displaystyle (X,{\mathcal {U}})}$ is called a complete uniform space (respectively, a sequentially complete uniform space) if every Cauchy prefilter (respectively, every elementary Cauchy prefilter) on ${\displaystyle X}$ converges to at least one point of ${\displaystyle X}$ when ${\displaystyle X}$ is endowed with the topology induced by ${\displaystyle {\mathcal {U}}.}$

Case of a topological vector space

If ${\displaystyle (X,\tau )}$ is a topological vector space then for any ${\displaystyle S\subseteq X}$ and ${\displaystyle x\in X,}$

$${\displaystyle \Delta _{X}(N)\cdot S=S+N\qquad {\text{ and }}\qquad \Delta _{X}(N)\cdot x=x+N,}$$

and the topology induced on ${\displaystyle X}$ by the canonical uniformity is the same as the topology that ${\displaystyle X}$ started with (that is, it is ${\displaystyle \tau }$).

Preliminaries: Complete pseudometric spaces

We review the basic notions related to the general theory of complete pseudometric spaces. Recall that every metric is a pseudometric and that a pseudometric ${\displaystyle p}$ is a metric if and only if ${\displaystyle p(x,y)=0}$ implies ${\displaystyle x=y.}$ Thus every metric space is a pseudometric space and a pseudometric space ${\displaystyle (X,p)}$ is a metric space if and only if ${\displaystyle p}$ is a metric.

If ${\displaystyle S}$ is a subset of a pseudometric space ${\displaystyle (X,d)}$ then the diameter of ${\displaystyle S}$ is defined to be

$${\displaystyle \operatorname {diam} (S)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\sup _{}\{d(s,t):s,t\in S\}.}$$

A prefilter ${\displaystyle {\mathcal {B}}}$ on a pseudometric space ${\displaystyle (X,d)}$ is called a ${\displaystyle d}$-Cauchy prefilter or simply a Cauchy prefilter if for each real ${\displaystyle r>0,}$ there is some ${\displaystyle B\in {\mathcal {B}}}$ such that the diameter of ${\displaystyle B}$ is less than ${\displaystyle r.}$

Suppose ${\displaystyle (X,d)}$ is a pseudometric space. A net ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}$ in ${\displaystyle X}$ is called a ${\displaystyle d}$-Cauchy net or simply a Cauchy net if ${\displaystyle \operatorname {Tails} \left(x_{\bullet }\right)}$ is a Cauchy prefilter, which happens if and only if

for every ${\displaystyle r>0}$ there is some ${\displaystyle i\in I}$ such that if ${\displaystyle j,k\in I}$ with ${\displaystyle j\geq i}$ and ${\displaystyle k\geq i}$ then ${\displaystyle d\left(x_{j},x_{k}\right)<r}$

or equivalently, if and only if ${\displaystyle \left(d\left(x_{j},x_{k}\right)\right)_{(i,j)\in I\times I}\to 0}$ in ${\displaystyle \mathbb {R} .}$ This is analogous to the following characterization of the converge of ${\displaystyle x_{\bullet }}$ to a point: if ${\displaystyle x\in X,}$ then ${\displaystyle x_{\bullet }\to x}$ in ${\displaystyle (X,d)}$ if and only if ${\displaystyle \left(x_{i},x\right)_{i\in I}\to 0}$ in ${\displaystyle \mathbb {R} .}$

A Cauchy sequence is a sequence that is also a Cauchy net.^{[note 3]}

Every pseudometric ${\displaystyle p}$ on a set ${\displaystyle X}$ induces the usual canonical topology on ${\displaystyle X,}$ which we'll denote by ${\displaystyle \tau _{p}}$; it also induces a canonical uniformity on ${\displaystyle X,}$ which we'll denote by ${\displaystyle {\mathcal {U}}_{p}.}$ The topology on ${\displaystyle X}$ induced by the uniformity ${\displaystyle {\mathcal {U}}_{p}}$ is equal to ${\displaystyle \tau _{p}.}$ A net ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}$ in ${\displaystyle X}$ is Cauchy with respect to ${\displaystyle p}$ if and only if it is Cauchy with respect to the uniformity ${\displaystyle {\mathcal {U}}_{p}.}$ The pseudometric space ${\displaystyle (X,p)}$ is a complete (resp. a sequentially complete) pseudometric space if and only if ${\displaystyle \left(X,{\mathcal {U}}_{p}\right)}$ is a complete (resp. a sequentially complete) uniform space. Moreover, the pseudometric space ${\displaystyle (X,p)}$ (resp. the uniform space ${\displaystyle \left(X,{\mathcal {U}}_{p}\right)}$) is complete if and only if it is sequentially complete.

A pseudometric space ${\displaystyle (X,d)}$ (for example, a metric space) is called complete and ${\displaystyle d}$ is called a complete pseudometric if any of the following equivalent conditions hold:

1. Every Cauchy prefilter on ${\displaystyle X}$ converges to at least one point of ${\displaystyle X.}$
2. The above statement but with the word "prefilter" replaced by "filter."
3. Every Cauchy net in ${\displaystyle X}$ converges to at least one point of ${\displaystyle X.}$
   • If ${\displaystyle d}$ is a metric on ${\displaystyle X}$ then any limit point is necessarily unique and the same is true for limits of Cauchy prefilters on ${\displaystyle X.}$
4. Every Cauchy sequence in ${\displaystyle X}$ converges to at least one point of ${\displaystyle X.}$
   • Thus to prove that ${\displaystyle (X,d)}$ is complete, it suffices to only consider Cauchy sequences in ${\displaystyle X}$ (and it is not necessary to consider the more general Cauchy nets).
5. The canonical uniformity on ${\displaystyle X}$ induced by the pseudometric ${\displaystyle d}$ is a complete uniformity.

And if addition ${\displaystyle d}$ is a metric then we may add to this list:

6. Every decreasing sequence of closed balls whose diameters shrink to ${\displaystyle 0}$ has non-empty intersection.^[8]

Complete pseudometrics and complete TVSs

Every F-space, and thus also every Fréchet space, Banach space, and Hilbert space is a complete TVS. Note that every F-space is a Baire space but there are normed spaces that are Baire but not Banach.^[9]

A pseudometric ${\displaystyle d}$ on a vector space ${\displaystyle X}$ is said to be a translation invariant pseudometric if ${\displaystyle d(x,y)=d(x+z,y+z)}$ for all vectors ${\displaystyle x,y,z\in X.}$

Suppose ${\displaystyle (X,\tau )}$ is pseudometrizable TVS (for example, a metrizable TVS) and that ${\displaystyle p}$ is any pseudometric on ${\displaystyle X}$ such that the topology on ${\displaystyle X}$ induced by ${\displaystyle p}$ is equal to ${\displaystyle \tau .}$ If ${\displaystyle p}$ is translation-invariant, then ${\displaystyle (X,\tau )}$ is a complete TVS if and only if ${\displaystyle (X,p)}$ is a complete pseudometric space.^[10] If ${\displaystyle p}$ is not translation-invariant, then may be possible for ${\displaystyle (X,\tau )}$ to be a complete TVS but ${\displaystyle (X,p)}$ to not be a complete pseudometric space^[10] (see this footnote^{[note 4]} for an example).^[10]

Complete norms and equivalent norms

Two norms on a vector space are called equivalent if and only if they induce the same topology.^[13] If ${\displaystyle p}$ and ${\displaystyle q}$ are two equivalent norms on a vector space ${\displaystyle X}$ then the normed space ${\displaystyle (X,p)}$ is a Banach space if and only if ${\displaystyle (X,q)}$ is a Banach space. See this footnote for an example of a continuous norm on a Banach space that is not equivalent to that Banach space's given norm.^{[note 6][13]} All norms on a finite-dimensional vector space are equivalent and every finite-dimensional normed space is a Banach space.^[14] Every Banach space is a complete TVS. A normed space is a Banach space (that is, its canonical norm-induced metric is complete) if and only if it is complete as a topological vector space.

Examples of completions

For example, the vector space consisting of scalar-valued simple functions ${\displaystyle f}$ for which ${\displaystyle |f|_{p}<\infty }$ (where this seminorm is defined in the usual way in terms of Lebesgue integration) becomes a seminormed space when endowed with this seminorm, which in turn makes it into both a pseudometric space and a non-Hausdorff non-complete TVS; any completion of this space is a non-Hausdorff complete seminormed space that when quotiented by the closure of its origin (so as to obtain a Hausdorff TVS) results in (a space linearly isometrically-isomorphic to) the usual complete Hausdorff ${\displaystyle L^{p}}$-space (endowed with the usual complete ${\displaystyle \|\cdot \|_{p}}$ norm).