coherent topology

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In topology, a coherent topology is a topology that is uniquely determined by a family of Subspace topology, subspaces. Loosely speaking, a topological space is coherent with a family of subspaces if it is a ''topological union'' of those subspaces. It is also sometimes called the weak topology generated by the family of subspaces, a notion that is quite different from the notion of a weak topology generated by a set of maps.

# Definition

Let $X$ be a topological space and let $C = \left\$ be a indexed family, family of subsets of $X$ each having the subspace topology. (Typically $C$ will be a Cover (topology), cover of $X$.) Then $X$ is said to be coherent with $C$ (or determined by $C$)$X$ is also said to have the weak topology generated by $C.$ This is a potentially confusing name since the adjectives and are used with opposite meanings by different authors. In modern usage the term is synonymous with initial topology and is synonymous with final topology. It is the final topology that is being discussed here. if the topology of $X$ is recovered as the one coming from the final topology coinduced by the inclusion maps $i_\alpha : C_\alpha \to X \qquad \alpha \in A.$ By definition, this is the finest topology on (the underlying set of) $X$ for which the inclusion maps are Continuous function (topology), continuous. $X$ is coherent with $C$ if either of the following two equivalent conditions holds: * A subset $U$ is Open set, open in $X$ if and only if $U \cap C_$ is open in $C_$ for each $\alpha \in A.$ * A subset $U$ is Closed set, closed in $X$ if and only if $U \cap C_$ is closed in $C_$ for each $\alpha \in A.$ Given a topological space $X$ and any family of subspaces $C$ there is a unique topology on (the underlying set of) $X$ that is coherent with $C.$ This topology will, in general, be Comparison of topologies, finer than the given topology on $X.$

# Examples

* A topological space $X$ is coherent with every open cover of $X.$ * A topological space $X$ is coherent with every Locally finite collection, locally finite closed cover of $X.$ * A discrete space is coherent with every family of subspaces (including the Empty set, empty family). * A topological space $X$ is coherent with a Partition (set theory), partition of $X$ if and only $X$ is homeomorphic to the Disjoint union (topology), disjoint union of the elements of the partition. * Finitely generated spaces are those determined by the family of all Finite topological space, finite subspaces. * Compactly generated spaces are those determined by the family of all Compact space, compact subspaces. * A CW complex $X$ is coherent with its family of $n$-skeletons $X_n.$

# Topological union

Let $\left\$ be a family of (not necessarily Disjoint set, disjoint) topological spaces such that the Induced topology, induced topologies agree on each Intersection (set theory), intersection $X_ \cap X_.$ Assume further that $X_ \cap X_$ is closed in $X_$ for each $\alpha, \beta \in A.$ Then the topological union $X$ is the set-theoretic union $X^ = \bigcup_ X_\alpha$ endowed with the final topology coinduced by the inclusion maps $i_\alpha : X_\alpha \to X^$. The inclusion maps will then be topological embeddings and $X$ will be coherent with the subspaces $\left\.$ Conversely, if $X$ is coherent with a family of subspaces $\left\$ that cover $X,$ then $X$ is homeomorphic to the topological union of the family $\left\.$ One can form the topological union of an arbitrary family of topological spaces as above, but if the topologies do not agree on the intersections then the inclusions will not necessarily be embeddings. One can also describe the topological union by means of the Disjoint union (topology), disjoint union. Specifically, if $X$ is a topological union of the family $\left\,$ then $X$ is homeomorphic to the Quotient space (topology), quotient of the disjoint union of the family $\left\$ by the equivalence relation $(x,\alpha) \sim (y,\beta) \Leftrightarrow x = y$ for all $\alpha, \beta \in A.$; that is, $X \cong \coprod_X_\alpha / \sim .$ If the spaces $\left\$ are all disjoint then the topological union is just the disjoint union. Assume now that the set A is Directed set, directed, in a way compatible with inclusion: $\alpha \leq \beta$ whenever $X_\alpha\subset X_$. Then there is a unique map from $\varinjlim X_\alpha$ to $X,$ which is in fact a homeomorphism. Here $\varinjlim X_\alpha$ is the Direct limit, direct (inductive) limit (Limit (category theory)#Colimits, colimit) of $\left\$ in the category Category of topological spaces, Top.

# Properties

Let $X$ be coherent with a family of subspaces $\left\.$ A map $f : X \to Y$ is Continuous function (topology), continuous if and only if the restrictions $f\big\vert_ : C_ \to Y\,$ are continuous for each $\alpha \in A.$ This universal property characterizes coherent topologies in the sense that a space $X$ is coherent with $C$ if and only if this property holds for all spaces $Y$ and all functions $f : X \to Y.$ Let $X$ be determined by a Cover (topology), cover $C = \left\.$ Then * If $C$ is a Refinement (topology), refinement of a cover $D,$ then $X$ is determined by $D.$ * If $D$ is a refinement of $C$ and each $C_$ is determined by the family of all $D_$ contained in $C_$ then $X$ is determined by $D.$ * Let $X$ be determined by $\left\$ and let $Y$ be an open or closed Subspace (topology), subspace of $X.$ Then $Y$ is determined by $\left\.$ * Let $X$ be determined by $\left\$ and let $f : X \to Y$ be a quotient map. Then $Y$ is determined by $\left\.$ Let $f : X \to Y$ be a surjective map and suppose $Y$ is determined by $\left\.$ For each $\alpha \in A$ let $f_\alpha : f^\left(D_\alpha\right) \to D_\alpha\,$be the restriction of $f$ to $f^\left\left(D_\right\right).$ Then * If $f$ is continuous and each $f_$ is a quotient map, then $f$ is a quotient map. * $f$ is a closed map (resp. open map) if and only if each $f_$ is closed (resp. open).