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Why is it important for a Homeomorphic Function (by the definition below) to be continuous ? What purpose does continuity serve ?

Let M and N be metric spaces. A function f : M $\to$ N is a homeomorphism if it is a bijection, and both f: M $\to$ N and its inverse $f^{-1}$: N $\to$ M are continuous. We say M and and N are homeomorphic.

(I am too new at this, so the answer might as well be, "because we defined it like that", but I thought I should ask.)

Mathnoob
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  • If there is a homeomorphism between $M$ and $N$ this means that $M$ and $N$ have roughly the same shape in the sense that we get from one to the other without tearing. If one of them has "a single hole" then the other one does as well, etc. This is what continuity guarantees. – Snaw Feb 17 '22 at 06:57
  • Metric spaces have a distance function. If a function is a homeomorphism then if two points are close together, so are their images (and also in the reverse direction). A plain bijection would be able to ignore distances altogether. – Peter Feb 17 '22 at 07:02
  • Thanks. I really liked the answer from @Snaw as it was very easy to understand. – Mathnoob Feb 17 '22 at 07:52

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When you have a Metric spaces, you have a notion of distance that is described by a metric. In particular, you have the idea of points being "near" each other.

By continuity, you get that a convergent sequence is mapped to a convergent sequence, so a continuous map maps points near one another to points that are nearby.

What a homeomorphism does is that it allows you to not only map continuously, but also go in the reverse direction. This ability helps as then the metrics become equivalent, that is, the notion of distance in the two metrics is very similar.

PCeltide
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