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An example sometimes used to differentiate between homeomorphisms and homotopy equivalences, for example in this answer by @JohnGowers and on the Wikipedia article for topology, is that of the letters $\mathsf{X}$ and $\mathsf{Y}$. However, what I'm confused about is whether these spaces are meant to be polygons or made up of lines. In other words, does $\mathsf{X}$ signify A or B in this figure? A or B

My understanding right now is that in the following figure, C and D are homotopy equivalent but not homeomorphic, while E and F are both homeomorphic and homotopy equivalent: enter image description here Is this correct?

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The links you cite are viewing the letters $X$, $Y$ etc. as made up of line segments, as in your Figures $A$, $C$ and $D$. (The proofs given wouldn't make sense otherwise: deleting a point from a connected polygon cannot disconnect it.)

Your understanding about the figures $C$, $D$, $E$ and $F$ is correct. They are all contractible and hence homotopy-equivalent. The equivalence classes up to homeomorphism are $\{C\}$, $\{D\}$ and $\{E, F\}$.

The above is taking the $E$ and $F$ to be the spaces that include the two-dimensional region enclosed by the blue lines. If $E'$ and $F'$ are the spaces that just comprise the blue lines, then $E'$ and $F'$ are homeomorphic, but they are not contractible and they are not homotopy-equivalent to $C$ and $D$.

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  • $\begingroup$ Your reasoning about the point was what I thought too, but I wanted to confirm it! $\endgroup$ Jan 28, 2017 at 19:16
  • $\begingroup$ Wait, are $C$ and $D$ homotopy equivalent to $E$ and $F$? $\endgroup$ Jan 28, 2017 at 21:09
  • $\begingroup$ See clarification above - it depends how you interpret your figures. $\endgroup$
    – Rob Arthan
    Jan 28, 2017 at 22:17
  • $\begingroup$ Ok, that makes sense. Thanks! $\endgroup$ Jan 29, 2017 at 0:41

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