Symbols for homeomorphic, isomorphic, homotopic

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Discussion Overview

The discussion revolves around the symbols used to denote homeomorphic, isomorphic, and homotopic relationships in mathematics. Participants explore the conventions and variations in usage across different texts and contexts.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant notes that their professor uses the symbols \approx, \cong, and \simeq interchangeably for homeomorphic, isomorphic, and homotopic, raising the question of official symbols.
  • Another participant suggests that there is no universal convention for homeomorphic and homotopic symbols, while indicating that \simeq is commonly used for isomorphic in group theory.
  • A different participant claims that isomorphic is more frequently represented by \cong, but acknowledges that \simeq is also seen, particularly for homotopic, alongside \sim.
  • One participant mentions that they have not encountered a text where symbols for these concepts were used without prior definitions.
  • A later reply reiterates the lack of standard convention, emphasizing that authors choose their preferred symbols for these concepts in their texts.

Areas of Agreement / Disagreement

Participants generally agree that there is no standard convention for the symbols representing homeomorphic and homotopic relationships, and multiple competing views on the usage of these symbols remain unresolved.

Contextual Notes

Limitations include the absence of universally accepted definitions for the symbols in question and the variability in usage across different mathematical texts.

quasar987
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My professor seems to randomly use the symbols \approx, \cong and \simeq to denote either homeomorphic, isomorphic or homotopic. So, what is the official symbol for each?
 
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I don't think there's a universal convention for the first and the third, but in group theory "isomorphic to" is denoted by \simeq.
 
Isomorphic is more frequently \cong, though the latex tag for it is cong. The symbol Dexter gave is actually in my experience likely to beused for homotopic, though I also see \sim a lot as well.
 
I have never used a text in which the author used a symbol for these concepts without first defining it.
 
No doubt, but the question is: which symbol did they use for each concept?
 
quasar987 said:
No doubt, but the question is: which symbol did they use for each concept?

That's what my post was meant to convey: there is no standard convention and so each author sets up his text to so that the symbol he likes most is used most frequently.
 

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