Is the intersection of two topological spaces, a topological space?

Question

Suppose $(X, \mathcal{T})$ and $(Y, \mathcal{K})$ are two arbitrary topological spaces, and we don't know if $X \subseteq Y$, or $Y \subseteq X$ or if $X \not\subseteq Y$ or $Y \not\subseteq X$ or if $X \cap Y = \emptyset$

Can we say if $X \cap Y$ is a topological space? If so how would we define the topology on $X \cap Y$? Would we just take $\mathcal{T} \cap \mathcal{K}$?

I ask this as I've never seen a case like this defined before for topological spaces, and I'm not sure why.

Answer 1

For spaces (X,T1), (Y,T2) a compatable topology for X cap Y is the smallest topology containg T1 cap T2. This construction is valid for any intersection of spaces. Notice when X and Y are disjoint the resulting space is the empty space, the unique space with exactly one open set.

Exercise. What is the compatable topology of the lower and upper limit topologies?

Is there any reason or use for compatable topologies?