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Is there a symbol for "homeomorphic to"? I looked on Wikipedia, but it doesn't seem to mention one? Also, for isomorphism, is the symbol a squiggly line over an equals sign? What is the symbol with a squiggly line over just one horizontal line? Thanks.

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By the way, the LaTeX command to produce$\qquad$ $\cong$ is \cong$\qquad$ $\simeq$ is \simeq$\qquad$ $\sim$ is \sim. To use a LaTeX command, one encloses it in dollar signs, e.g. $\simeq$. If you see some math on this website, and you want to know the LaTeX code that produced it, you can right click on it and choose "Show Source". There is also an SE website devoted to LaTeX. – Zev Chonoles Sep 28 '11 at 16:30
Ah thanks! That will sure come in handy. – james Sep 28 '11 at 16:48
up vote 4 down vote accepted

I use $\cong$ for isomorphism in a category, which includes both homeomorphism and isomorphism of groups, etc. I have seen $\simeq$ used to mean homotopy equivalence, but I don't know how standard this is.

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I agree with your usage, but I do have a professor who writes his isomorphisms with $\simeq$ (which I adopted for a while, then thought better of it). – Zev Chonoles Sep 28 '11 at 16:25
I believe it's quite common to use $\approx$ for homeomorphisms (or perhaps isomorphisms with a more "geometric" flavor, like isometries etc.) – Miha Habič Sep 28 '11 at 16:56

I agree with Qiaochu Yuan's answer for the most part, but if you are working in an area where you must distinguish between homeomorphism, homotopy equivalence, and diffeomorphism, the standard notation becomes ambiguous. This has been relevant for me because I've been studying the interplay of topology and geometry for hyperbolic 3-manifolds. In this context what seems to be the most consistent is $\sim$ for homotopy equivalence, $\simeq$ for homeomorphic, and $\approx$ for diffeomorphic. This way $\sim$ agrees with usage as indicating same members of an equivalence class, where the equivalence class is that of the fundamental group; and $\approx$ agrees with what geometers like (for instance John Lee), and $\simeq$ is just a decent choice for something that looks halfway between them.

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Since $\simeq$ is nearly universally used for both homotopic and homotopy equivalent, using it for homeomorphic is going to confuse a lot of readers. In my books, I've used $\approx$ for both homeomorphic and diffeomorphic, depending on context. The symbol $\cong$ can in principle be used to designate an isomorphism in any category (e.g., isometric, diffeomorphic, homeomorphic, linearly isomorphic, etc.). The bottom line is that you have to specify what you mean by any such symbol. – Jack Lee Dec 14 '13 at 1:02
Sorry to put words in your mouth! I guess it does depend a lot on context. I was wondering though, let's say for some reason it was very important to distinguish objects that were homeomorphic but not diffeomorphic, and you absolutely had to choose different symbols. What would you choose? (BTW I've read both of your last 2 books and respect your opinion a lot!) – j0equ1nn Dec 28 '13 at 10:50
That's a tough one. I think my first inclination would be to choose one of the relations to designate with $\approx$, and then use words to designate the other. Alternatively, I might use $\approx$ for homeomorphic and $\cong$ for diffeomorphic, provided I didn't need the latter symbol for something else. – Jack Lee Dec 28 '13 at 16:07
Oh, and many thanks for the compliment, @JoeQuinn! – Jack Lee Dec 28 '13 at 16:07

It depends heavily on the field; in the set theoretic literature I'm familiar with (that is, the odd corner of the literature related to NF) I most often see $\sim$ for an arbitrary equivalence relation, and $\simeq$ for isomorphism. (But this is also literature that held on to $\hat{x}(\phi)$ instead of the modern $\{x:\phi\}$ for quite some time...) I have also seen some category theory authors use $\simeq$ for isomorphisms in a category, while $\cong$ is reserved specifically for natural isomorphisms between functors.

In general, I think $\cong$ is most likely to be recognized as isomorphism in the abstract sense of being an invertible morphism of some kind, but I generally count on having to get used to any given author using a different notation. It definitely never hurts to establish your usage explicitly if you're going to be writing a mathematical document.

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