Characteristic property

The initial topology on ${\displaystyle X}$ can be characterized by the following characteristic property:
A function ${\displaystyle g}$ from some space ${\displaystyle Z}$ to ${\displaystyle X}$ is continuous if and only if ${\displaystyle f_{i}\circ g}$ is continuous for each ${\displaystyle i\in I.}$^[4]

Characteristic property of the initial topology

Note that, despite looking quite similar, this is not a universal property. A categorical description is given below.

A filter ${\displaystyle {\mathcal {B}}}$ on ${\displaystyle X}$ converges to a point ${\displaystyle x\in X}$ if and only if the prefilter ${\displaystyle f_{i}({\mathcal {B}})}$ converges to ${\displaystyle f_{i}(x)}$ for every ${\displaystyle i\in I.}$^[4]

Evaluation

By the universal property of the product topology, we know that any family of continuous maps ${\displaystyle f_{i}:X\to Y_{i}}$ determines a unique continuous map

$${\displaystyle {\begin{alignedat}{4}f:\;&&X&&\;\to \;&\prod _{i}Y_{i}\\[0.3ex]&&x&&\;\mapsto \;&\left(f_{i}(x)\right)_{i\in I}\\\end{alignedat}}}$$

This map is known as the evaluation map.

A family of maps ${\displaystyle \{f_{i}:X\to Y_{i}\}}$ is said to separate points in ${\displaystyle X}$ if for all ${\displaystyle x\neq y}$ in ${\displaystyle X}$ there exists some ${\displaystyle i}$ such that ${\displaystyle f_{i}(x)\neq f_{i}(y).}$ The family ${\displaystyle \{f_{i}\}}$ separates points if and only if the associated evaluation map ${\displaystyle f}$ is injective.

The evaluation map ${\displaystyle f}$ will be a topological embedding if and only if ${\displaystyle X}$ has the initial topology determined by the maps ${\displaystyle \{f_{i}\}}$ and this family of maps separates points in ${\displaystyle X.}$

Hausdorffness

If ${\displaystyle X}$ has the initial topology induced by ${\displaystyle \left\{f_{i}:X\to Y_{i}\right\}}$ and if every ${\displaystyle Y_{i}}$ is Hausdorff, then ${\displaystyle X}$ is a Hausdorff space if and only if these maps separate points on ${\displaystyle X.}$^[3]

Transitivity of the initial topology

If ${\displaystyle X}$ has the initial topology induced by the ${\displaystyle I}$-indexed family of mappings ${\displaystyle \left\{f_{i}:X\to Y_{i}\right\}}$ and if for every ${\displaystyle i\in I,}$ the topology on ${\displaystyle Y_{i}}$ is the initial topology induced by some ${\displaystyle J_{i}}$-indexed family of mappings ${\displaystyle \left\{g_{j}:Y_{i}\to Z_{j}\right\}}$ (as ${\displaystyle j}$ ranges over ${\displaystyle J_{i}}$), then the initial topology on ${\displaystyle X}$ induced by ${\displaystyle \left\{f_{i}:X\to Y_{i}\right\}}$ is equal to the initial topology induced by the ${\displaystyle {\textstyle \bigcup \limits _{i\in I}J_{i}}}$-indexed family of mappings ${\displaystyle \left\{g_{j}\circ f_{i}:X\to Z_{j}\right\}}$ as ${\displaystyle i}$ ranges over ${\displaystyle I}$ and ${\displaystyle j}$ ranges over ${\displaystyle J_{i}.}$^[5] Several important corollaries of this fact are now given.

In particular, if ${\displaystyle S\subseteq X}$ then the subspace topology that ${\displaystyle S}$ inherits from ${\displaystyle X}$ is equal to the initial topology induced by the inclusion map ${\displaystyle S\to X}$ (defined by ${\displaystyle s\mapsto s}$). Consequently, if ${\displaystyle X}$ has the initial topology induced by ${\displaystyle \left\{f_{i}:X\to Y_{i}\right\}}$ then the subspace topology that ${\displaystyle S}$ inherits from ${\displaystyle X}$ is equal to the initial topology induced on ${\displaystyle S}$ by the restrictions ${\displaystyle \left\{\left.f_{i}\right|_{S}:S\to Y_{i}\right\}}$ of the ${\displaystyle f_{i}}$ to ${\displaystyle S.}$^[4]

The product topology on ${\displaystyle \prod _{i}Y_{i}}$ is equal to the initial topology induced by the canonical projections ${\displaystyle \operatorname {pr} _{i}:\left(x_{k}\right)_{k\in I}\mapsto x_{i}}$ as ${\displaystyle i}$ ranges over ${\displaystyle I.}$^[4] Consequently, the initial topology on ${\displaystyle X}$ induced by ${\displaystyle \left\{f_{i}:X\to Y_{i}\right\}}$ is equal to the inverse image of the product topology on ${\displaystyle \prod _{i}Y_{i}}$ by the evaluation map ${\textstyle f:X\to \prod _{i}Y_{i}\,.}$^[4] Furthermore, if the maps ${\displaystyle \left\{f_{i}\right\}_{i\in I}}$ separate points on ${\displaystyle X}$ then the evaluation map is a homeomorphism onto the subspace ${\displaystyle f(X)}$ of the product space ${\displaystyle \prod _{i}Y_{i}.}$^[4]

Separating points from closed sets

If a space ${\displaystyle X}$ comes equipped with a topology, it is often useful to know whether or not the topology on ${\displaystyle X}$ is the initial topology induced by some family of maps on ${\displaystyle X.}$ This section gives a sufficient (but not necessary) condition.

A family of maps ${\displaystyle \left\{f_{i}:X\to Y_{i}\right\}}$ separates points from closed sets in ${\displaystyle X}$ if for all closed sets ${\displaystyle A}$ in ${\displaystyle X}$ and all ${\displaystyle x\not \in A,}$ there exists some ${\displaystyle i}$ such that

$${\displaystyle f_{i}(x)\notin \operatorname {cl} (f_{i}(A))}$$

where ${\displaystyle \operatorname {cl} }$ denotes the closure operator.

Theorem. A family of continuous maps ${\displaystyle \left\{f_{i}:X\to Y_{i}\right\}}$ separates points from closed sets if and only if the cylinder sets ${\displaystyle f_{i}^{-1}(V),}$ for ${\displaystyle V}$ open in ${\displaystyle Y_{i},}$ form a base for the topology on ${\displaystyle X.}$

It follows that whenever ${\displaystyle \left\{f_{i}\right\}}$ separates points from closed sets, the space ${\displaystyle X}$ has the initial topology induced by the maps ${\displaystyle \left\{f_{i}\right\}.}$ The converse fails, since generally the cylinder sets will only form a subbase (and not a base) for the initial topology.

If the space ${\displaystyle X}$ is a T₀ space, then any collection of maps ${\displaystyle \left\{f_{i}\right\}}$ that separates points from closed sets in ${\displaystyle X}$ must also separate points. In this case, the evaluation map will be an embedding.

Initial uniform structure

If ${\displaystyle \left({\mathcal {U}}_{i}\right)_{i\in I}}$ is a family of uniform structures on ${\displaystyle X}$ indexed by ${\displaystyle I\neq \varnothing ,}$ then the least upper bound uniform structure of ${\displaystyle \left({\mathcal {U}}_{i}\right)_{i\in I}}$ is the coarsest uniform structure on ${\displaystyle X}$ that is finer than each ${\displaystyle {\mathcal {U}}_{i}.}$ This uniform always exists and it is equal to the filter on ${\displaystyle X\times X}$ generated by the filter subbase ${\displaystyle {\textstyle \bigcup \limits _{i\in I}{\mathcal {U}}_{i}}.}$^[6] If ${\displaystyle \tau _{i}}$ is the topology on ${\displaystyle X}$ induced by the uniform structure ${\displaystyle {\mathcal {U}}_{i}}$ then the topology on ${\displaystyle X}$ associated with least upper bound uniform structure is equal to the least upper bound topology of ${\displaystyle \left(\tau _{i}\right)_{i\in I}.}$^[6]

Now suppose that ${\displaystyle \left\{f_{i}:X\to Y_{i}\right\}}$ is a family of maps and for every ${\displaystyle i\in I,}$ let ${\displaystyle {\mathcal {U}}_{i}}$ be a uniform structure on ${\displaystyle Y_{i}.}$ Then the initial uniform structure of the ${\displaystyle Y_{i}}$ by the mappings ${\displaystyle f_{i}}$ is the unique coarsest uniform structure ${\displaystyle {\mathcal {U}}}$ on ${\displaystyle X}$ making all ${\displaystyle f_{i}:\left(X,{\mathcal {U}}\right)\to \left(Y_{i},{\mathcal {U}}_{i}\right)}$ uniformly continuous.^[6] It is equal to the least upper bound uniform structure of the ${\displaystyle I}$-indexed family of uniform structures ${\displaystyle f_{i}^{-1}\left({\mathcal {U}}_{i}\right)}$ (for ${\displaystyle i\in I}$).^[6] The topology on ${\displaystyle X}$ induced by ${\displaystyle {\mathcal {U}}}$ is the coarsest topology on ${\displaystyle X}$ such that every ${\displaystyle f_{i}:X\to Y_{i}}$ is continuous.^[6] The initial uniform structure ${\displaystyle {\mathcal {U}}}$ is also equal to the coarsest uniform structure such that the identity mappings ${\displaystyle \operatorname {id}$ :\left(X,{\mathcal {U}}\right)\to \left(X,f_{i}^{-1}\left({\mathcal {U}}_{i}\right)\right)} are uniformly continuous.^[6]

Hausdorffness: The topology on ${\displaystyle X}$ induced by the initial uniform structure ${\displaystyle {\mathcal {U}}}$ is Hausdorff if and only if for whenever ${\displaystyle x,y\in X}$ are distinct (${\displaystyle x\neq y}$) then there exists some ${\displaystyle i\in I}$ and some entourage ${\displaystyle V_{i}\in {\mathcal {U}}_{i}}$ of ${\displaystyle Y_{i}}$ such that ${\displaystyle \left(f_{i}(x),f_{i}(y)\right)\not \in V_{i}.}$^[6] Furthermore, if for every index ${\displaystyle i\in I,}$ the topology on ${\displaystyle Y_{i}}$ induced by ${\displaystyle {\mathcal {U}}_{i}}$ is Hausdorff then the topology on ${\displaystyle X}$ induced by the initial uniform structure ${\displaystyle {\mathcal {U}}}$ is Hausdorff if and only if the maps ${\displaystyle \left\{f_{i}:X\to Y_{i}\right\}}$ separate points on ${\displaystyle X}$^[6] (or equivalently, if and only if the evaluation map ${\textstyle f:X\to \prod _{i}Y_{i}}$ is injective)

Uniform continuity: If ${\displaystyle {\mathcal {U}}}$ is the initial uniform structure induced by the mappings ${\displaystyle \left\{f_{i}:X\to Y_{i}\right\},}$ then a function ${\displaystyle g}$ from some uniform space ${\displaystyle Z}$ into ${\displaystyle (X,{\mathcal {U}})}$ is uniformly continuous if and only if ${\displaystyle f_{i}\circ g:Z\to Y_{i}}$ is uniformly continuous for each ${\displaystyle i\in I.}$^[6]

Cauchy filter: A filter ${\displaystyle {\mathcal {B}}}$ on ${\displaystyle X}$ is a Cauchy filter on ${\displaystyle (X,{\mathcal {U}})}$ if and only if ${\displaystyle f_{i}\left({\mathcal {B}}\right)}$ is a Cauchy prefilter on ${\displaystyle Y_{i}}$ for every ${\displaystyle i\in I.}$^[6]

Transitivity of the initial uniform structure: If the word "topology" is replaced with "uniform structure" in the statement of "transitivity of the initial topology" given above, then the resulting statement will also be true.