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Let $X$ and $Y$ be topological spaces, and let $A\subseteq X$ and $B\subseteq Y$ be subsets. Then we naturally give $A$ and $B$ the (subspace) topologies induced by the inclusion maps $$i : A\hookrightarrow X\qquad\qquad\text{and}\qquad\qquad j : B\hookrightarrow Y.$$

What about the product $A\times B$? There are two natural options for topologizing $A\times B$, namely:

  • $A\times B$ is the product of the spaces $A$ and $B$, so you give it the product topology
  • first give $X\times Y$ the product topology and then observe that the underlying set of $A\times B$ is a subset of $X\times Y$, so it makes sense to give $A\times B$ the (subspace) topology induced by the inclusion map $k : A\times B\to X\times Y$.

Are these two topologies on $A\times B$ the same?

Could anyone give me a hint on how to solve this problem?

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Emptymind
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2 Answers2

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I gave the general argument (for all products, even the box product) here in detail.

A more general view, based on the fact that both the subspace and the product topology are special cases of initial topologies can be found here; the result you're looking for is at the end. The whole theory outlined is standard "categorical topology", one could say.

Henno Brandsma
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You can show identity between two space (same sets, different(really?) topology.) is isomorphism.

try to choose same bases of two spaces.

Hint: $$ (U \times V) \cap (A \times B) = (U \cap A) \times (V \cap B) $$

If you are in big trouble, you can find answer in introduction to topology by munkres, Thm 16.3.

Needmoremath
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