Notations involving squiggly lines over horizontal lines 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. 
 A: 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.
A: 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. 
A: 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.
