# What is the difference between isomorphism and homeomorphism?

I have some questions understanding isomorphism. Wikipedia said that

isomorphism is bijective homeomorphism

I kown that $F$ is a homeomorphism if $F$ and $F^{-1}$ are continuous. So my question is: If $F$ and its inverse are continuous, can it not be bijective? Any example? I think if $F$ and its inverse are both continuous, they ought to be bijective, is that right?

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You mixed up two structures: homeomorphism and homomorphism. Isomorphism is a bijective HOMOMORPHISM. – Bartek Pawlik Jun 14 '13 at 8:45
@Bartek: No. For some structures, the isomorphisms are the bijective homomorphisms; but the correct definition is "homomorphism that has an inverse homomorphism". – Zev Chonoles Jun 14 '13 at 8:52
1. Homeomorphism is bijective in nature. 2. A map has to be bijective to have a (two-sided) inverse. 3. A one-sided inverse couple need not be bijective. It is justified by any retract of topological spaces. – Anonymous Coward Jun 14 '13 at 8:57

When talking about functions between sets, there is no such thing as an "inverse" in the first place if the map is not bijective. Take a look at the relevant Wikipedia page.

Moreover, when you refer to

isomorphism is bijective homeomorphism

I think you're thinking of

isomorphism is bijective homomorphism

which, by the way, happens to be true in some nice cases (groups, rings, etc.), but is absolutely not the definition of "isomorphism" in general, and in particular, is not true for topological spaces - there are functions $f:X\to Y$ that are bijective and continuous, but are not homeomorphisms.

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Isomorphism and homeomorphism appear both in topology and abstract algebra. Here, I think you mean isomorphism and homeomorphism in topology. In one word, they are the same in topology.

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