TITLE: Notations involving squiggly lines over horizontal lines
QUESTION [18 upvotes]: 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.

REPLY [2 votes]: 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.