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\; =\; \backslash left\backslash $ 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 $$i\_\backslash alpha\; :\; C\_\backslash alpha\; \backslash to\; X\; \backslash qquad\; \backslash alpha\; \backslash in\; A.$$ 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 $U\; \backslash cap\; C\_$ is open in $C\_$ for each $\backslash alpha\; \backslash in\; A.$ * A subset $U$ is Closed set, closed in $X$ if and only if $U\; \backslash cap\; C\_$ is closed in $C\_$ for each $\backslash alpha\; \backslash in\; A.$ 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 $\backslash left\backslash $ 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 $X\_\; \backslash cap\; X\_.$ Assume further that $X\_\; \backslash cap\; X\_$ is closed in $X\_$ for each $\backslash alpha,\; \backslash beta\; \backslash in\; A.$ Then the topological union $X$ is the set-theoretic union $$X^\; =\; \backslash bigcup\_\; X\_\backslash alpha$$ endowed with the final topology coinduced by the inclusion maps $i\_\backslash alpha\; :\; X\_\backslash alpha\; \backslash to\; X^$. The inclusion maps will then be topological embeddings and $X$ will be coherent with the subspaces $\backslash left\backslash .$ Conversely, if $X$ is coherent with a family of subspaces $\backslash left\backslash $ that cover $X,$ then $X$ is homeomorphic to the topological union of the family $\backslash left\backslash .$ 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 $\backslash left\backslash ,$ then $X$ is homeomorphic to the Quotient space (topology), quotient of the disjoint union of the family $\backslash left\backslash $ by the equivalence relation $$(x,\backslash alpha)\; \backslash sim\; (y,\backslash beta)\; \backslash Leftrightarrow\; x\; =\; y$$ for all $\backslash alpha,\; \backslash beta\; \backslash in\; A.$; that is, $$X\; \backslash cong\; \backslash coprod\_X\_\backslash alpha\; /\; \backslash sim\; .$$ If the spaces $\backslash left\backslash $ 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: $\backslash alpha\; \backslash leq\; \backslash beta$ whenever $X\_\backslash alpha\backslash subset\; X\_$. Then there is a unique map from $\backslash varinjlim\; X\_\backslash alpha$ to $X,$ which is in fact a homeomorphism. Here $\backslash varinjlim\; X\_\backslash alpha$ is the Direct limit, direct (inductive) limit (Limit (category theory)#Colimits, colimit) of $\backslash left\backslash $ in the category Category of topological spaces, Top.Properties

Let $X$ be coherent with a family of subspaces $\backslash left\backslash .$ A map $f\; :\; X\; \backslash to\; Y$ is Continuous function (topology), continuous if and only if the restrictions $$f\backslash big\backslash vert\_\; :\; C\_\; \backslash to\; Y\backslash ,$$ are continuous for each $\backslash alpha\; \backslash in\; A.$ 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 $f\; :\; X\; \backslash to\; Y.$ Let $X$ be determined by a Cover (topology), cover $C\; =\; \backslash left\backslash .$ 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 $\backslash left\backslash $ and let $Y$ be an open or closed Subspace (topology), subspace of $X.$ Then $Y$ is determined by $\backslash left\backslash .$ * Let $X$ be determined by $\backslash left\backslash $ and let $f\; :\; X\; \backslash to\; Y$ be a quotient map. Then $Y$ is determined by $\backslash left\backslash .$ Let $f\; :\; X\; \backslash to\; Y$ be a surjective map and suppose $Y$ is determined by $\backslash left\backslash .$ For each $\backslash alpha\; \backslash in\; A$ let $f\_\backslash alpha\; :\; f^\backslash left(D\_\backslash alpha\backslash right)\; \backslash to\; D\_\backslash alpha\backslash ,$be the restriction of $f$ to $f^\backslash left(D\_\backslash right).$ 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).See also

*Notes

References

* * {{cite book, last=Willard, first=Stephen, title=General Topology, url=https://archive.org/details/generaltopology00will_0, url-access=registration, publisher=Addison-Wesley, location=Reading, Massachusetts, year=1970, isbn=0-486-43479-6, id=(Dover edition) General topology