In topology, a coherent topology is a topology that is uniquely determined by a family of 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.

Let ${\displaystyle X}$ be a topological space and let ${\displaystyle C=\left\{C_{\alpha }:\alpha \in A\right\}}$ be a family of subsets of ${\displaystyle X,}$ each with its induced subspace topology. (Typically ${\displaystyle C}$ will be a cover of ${\displaystyle X}$.) Then ${\displaystyle X}$ is said to be coherent with ${\displaystyle C}$ (or determined by ${\displaystyle C}$) if the topology of ${\displaystyle X}$ is recovered as the one coming from the final topology coinduced by the inclusion maps $${\displaystyle i_{\alpha }:C_{\alpha }\to X\qquad \alpha \in A.}$$ By definition, this is the finest topology on (the underlying set of) ${\displaystyle X}$ for which the inclusion maps are continuous. ${\displaystyle X}$ is coherent with ${\displaystyle C}$ if either of the following two equivalent conditions holds:

Let ${\displaystyle \left\{X_{\alpha }:\alpha \in A\right\}}$ be a family of (not necessarily disjoint) topological spaces such that the induced topologies agree on each intersection ${\displaystyle X_{\alpha }\cap X_{\beta }.}$ Assume further that ${\displaystyle X_{\alpha }\cap X_{\beta }}$ is closed in ${\displaystyle X_{\alpha }}$ for each ${\displaystyle \alpha ,\beta \in A.}$ Then the topological union ${\displaystyle X}$ is the set-theoretic union $${\displaystyle X^{set}=\bigcup _{\alpha \in A}X_{\alpha }}$$ endowed with the final topology coinduced by the inclusion maps ${\displaystyle i_{\alpha }:X_{\alpha }\to X^{set}}$. The inclusion maps will then be topological embeddings and ${\displaystyle X}$ will be coherent with the subspaces ${\displaystyle \left\{X_{\alpha }\right\}.}$

Conversely, if ${\displaystyle X}$ is a topological space and is coherent with a family of subspaces ${\displaystyle \left\{C_{\alpha }\right\}}$ that cover ${\displaystyle X,}$ then ${\displaystyle X}$ is homeomorphic to the topological union of the family ${\displaystyle \left\{C_{\alpha }\right\}.}$

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. Specifically, if ${\displaystyle X}$ is a topological union of the family ${\displaystyle \left\{X_{\alpha }\right\},}$ then ${\displaystyle X}$ is homeomorphic to the quotient of the disjoint union of the family ${\displaystyle \left\{X_{\alpha }\right\}}$ by the equivalence relation $${\displaystyle (x,\alpha )\sim (y,\beta )\Leftrightarrow x=y}$$ for all ${\displaystyle \alpha ,\beta \in A.}$; that is, $${\displaystyle X\cong \coprod _{\alpha \in A}X_{\alpha }/\sim .}$$

If the spaces ${\displaystyle \left\{X_{\alpha }\right\}}$ are all disjoint then the topological union is just the disjoint union.

Assume now that the set A is directed, in a way compatible with inclusion: ${\displaystyle \alpha \leq \beta }$ whenever ${\displaystyle X_{\alpha }\subset X_{\beta }}$. Then there is a unique map from ${\displaystyle \varinjlim X_{\alpha }}$ to ${\displaystyle X,}$ which is in fact a homeomorphism. Here ${\displaystyle \varinjlim X_{\alpha }}$ is the direct (inductive) limit (colimit) of ${\displaystyle \left\{X_{\alpha }\right\}}$ in the category Top.