[Math] What does “isomorphism” exactly refer to in topology

general-topologyreal-analysis

I am reading an article and I came across the sentence that states as below:

"A linear separating isomorphism from C(T) onto C(S) is continuous, in which C(S) and C(T) denote sup-normed Banach spaces of real or complex-valued continuous
functions on the compact Hausdorff spaces S and T, respectively."

The writer has used only the injective and surjective property to prove the theorem, so it has made me think that the isomorphism mentioned above is the same as a bijection map.
On the other hand, I have been used to seeing isomorphisms in Algebra. Is there anything more than this with the word isomorphism in the sentence above?

Any help would be highly appreciated.

Best Answer

Well isomorphism in the topology (or more correctly in the category of topological spaces) are homeomorphisms. That is if we have two topological spaces $X$ and $Y$, then saying that $X$ and $Y$ are isomorphic is exactly the same as saying $X$ and $Y$ are homeomorphic.

Now since we are working with Banach spaces (which are topological spaces themselves), there is a theorem in Functional Analysis called the open mapping theorem:

Open Mapping Theorem: If $X$ and $Y$ are Banach Spaces, then any continuous surjective linear map $L : X \to Y$ is an open map.

Now if we have the added property that $L$ is an injective map, which is what it seems like the author is asserting, then we obtain a continous bijective open map, from the open mapping theorem.

Now going back to topology, there is a nice lemma that one can prove.

Lemma: Let $X$ and $Y$ be two topological spaces. If $f : X \to Y$ is a continuous, bijective open map, then $f$ is a homeomorphism.

Putting all this together one can see that this lemma along with the open mapping theorem shows that we have an homeomorphism.

$^*$For sets, isomorphisms are bijections.

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