Characteristic property The initial topology on X can be characterized by the following characteristic property: A function g from some space Z to X is continuous if and only if f_i \circ g is continuous for each i \in I. Note that, despite looking quite similar, this is not a
universal property. A categorical description is given below. A
filter \mathcal{B} on X
converges to a point x \in X if and only if the
prefilter f_i(\mathcal{B})
converges to f_i(x) for every i \in I.
Evaluation By the universal property of the
product topology, we know that any family of continuous maps f_i : X \to Y_i determines a unique continuous map \begin{alignat}{4} f :\;&& X &&\;\to \;& \prod_i Y_i \\[0.3ex] && x &&\;\mapsto\;& \left(f_i(x)\right)_{i \in I} \\ \end{alignat} This map is known as the ''''''. A family of maps \{f_i : X \to Y_i\} is said to
Separating set| in X if for all x \neq y in X there exists some i such that f_i(x) \neq f_i(y). The family \{f_i\} separates points if and only if the associated evaluation map f is
injective. The evaluation map f will be a
topological embedding if and only if X has the initial topology determined by the maps \{f_i\} and this family of maps separates points in X.
Hausdorffness If X has the initial topology induced by \left\{f_i : X \to Y_i\right\} and if every Y_i is Hausdorff, then X is a
Hausdorff space if and only if these maps
separate points on X.
Transitivity of the initial topology If X has the initial topology induced by the I-indexed family of mappings \left\{f_i : X \to Y_i\right\} and if for every i \in I, the topology on Y_i is the initial topology induced by some J_i-indexed family of mappings \left\{g_j : Y_i \to Z_j\right\} (as j ranges over J_i), then the initial topology on X induced by \left\{f_i : X \to Y_i\right\} is equal to the initial topology induced by the {\textstyle \bigcup\limits_{i \in I} J_i}-indexed family of mappings \left\{g_j \circ f_i : X \to Z_j\right\} as i ranges over I and j ranges over J_i. Several important corollaries of this fact are now given. In particular, if S \subseteq X then the subspace topology that S inherits from X is equal to the initial topology induced by the
inclusion map S \to X (defined by s \mapsto s). Consequently, if X has the initial topology induced by \left\{f_i : X \to Y_i\right\} then the subspace topology that S inherits from X is equal to the initial topology induced on S by the restrictions \left\{\left.f_i\right|_S : S \to Y_i\right\} of the f_i to S. The
product topology on \prod_i Y_i is equal to the initial topology induced by the canonical projections \operatorname{pr}_i : \left(x_k\right)_{k \in I} \mapsto x_i as i ranges over I. Consequently, the initial topology on X induced by \left\{f_i : X \to Y_i\right\} is equal to the inverse image of the product topology on \prod_i Y_i by the
evaluation map f : X \to \prod_i Y_i\,. Furthermore, if the maps \left\{f_i\right\}_{i \in I}
separate points on X then the evaluation map is a
homeomorphism onto the subspace f(X) of the product space \prod_i Y_i.
Separating points from closed sets If a space X comes equipped with a topology, it is often useful to know whether or not the topology on X is the initial topology induced by some family of maps on X. This section gives a sufficient (but not necessary) condition. A family of maps \left\{f_i : X \to Y_i\right\}
separates points from closed sets in X if for all
closed sets A in X and all x \not\in A, there exists some i such that f_i(x) \notin \operatorname{cl}(f_i(A)) where \operatorname{cl} denotes the
closure operator. :
Theorem. A family of continuous maps \left\{f_i : X \to Y_i\right\} separates points from closed sets if and only if the cylinder sets f_i^{-1}(V), for V open in Y_i, form a
base for the topology on X. It follows that whenever \left\{f_i\right\} separates points from closed sets, the space X has the initial topology induced by the maps \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 X is a
T0 space, then any collection of maps \left\{f_i\right\} that separates points from closed sets in X must also separate points. In this case, the evaluation map will be an embedding.
Initial uniform structure If \left(\mathcal{U}_i\right)_{i \in I} is a family of
uniform structures on X indexed by I \neq \varnothing, then the of \left(\mathcal{U}_i\right)_{i \in I} is the coarsest uniform structure on X that is finer than each \mathcal{U}_i. This uniform always exists and it is equal to the
filter on X \times X generated by the
filter subbase {\textstyle \bigcup\limits_{i \in I} \mathcal{U}_i}. If \tau_i is the topology on X induced by the uniform structure \mathcal{U}_i then the topology on X associated with least upper bound uniform structure is equal to the least upper bound topology of \left(\tau_i\right)_{i \in I}. Now suppose that \left\{f_i : X \to Y_i\right\} is a family of maps and for every i \in I, let \mathcal{U}_i be a uniform structure on Y_i. Then the is the unique coarsest uniform structure \mathcal{U} on X making all f_i : \left(X, \mathcal{U}\right) \to \left(Y_i, \mathcal{U}_i\right)
uniformly continuous. It is equal to the least upper bound uniform structure of the I-indexed family of uniform structures f_i^{-1}\left(\mathcal{U}_i\right) (for i \in I). The topology on X induced by \mathcal{U} is the coarsest topology on X such that every f_i : X \to Y_i is continuous. The initial uniform structure \mathcal{U} is also equal to the coarsest uniform structure such that the identity mappings \operatorname{id} : \left(X, \mathcal{U}\right) \to \left(X, f_i^{-1}\left(\mathcal{U}_i\right)\right) are uniformly continuous.
Hausdorffness: The topology on X induced by the initial uniform structure \mathcal{U} is
Hausdorff if and only if for whenever x, y \in X are distinct (x \neq y) then there exists some i \in I and some entourage V_i \in \mathcal{U}_i of Y_i such that \left(f_i(x), f_i(y)\right) \not\in V_i. Furthermore, if for every index i \in I, the topology on Y_i induced by \mathcal{U}_i is Hausdorff then the topology on X induced by the initial uniform structure \mathcal{U} is Hausdorff if and only if the maps \left\{f_i : X \to Y_i\right\}
separate points on X (or equivalently, if and only if the
evaluation map f : X \to \prod_i Y_i is injective)
Uniform continuity: If \mathcal{U} is the initial uniform structure induced by the mappings \left\{f_i : X \to Y_i\right\}, then a function g from some uniform space Z into (X, \mathcal{U}) is
uniformly continuous if and only if f_i \circ g : Z \to Y_i is uniformly continuous for each i \in I.
Cauchy filter: A
filter \mathcal{B} on X is a
Cauchy filter on (X, \mathcal{U}) if and only if f_i\left(\mathcal{B}\right) is a Cauchy prefilter on Y_i for every i \in I.
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. ==Categorical description==