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The problem is asking to list half a dozen topological properties that aren't preserved under Homotopy.

I can only think of cardinality (contractible spaces), compactness($\Bbb R^n$ is contractible), and interval type (open vs closed both contractible) but I'm struggling to find three other examples.

Anyone have any ideas?

Oliver G
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4 Answers4

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Dimension is a very important topological invariant which is not preserved under homotopy.

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    As in a cylinder is homotopic to a circle, yet they're not homeomorphic? – Oliver G Apr 14 '17 at 21:24
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    Exactly, or more simply $\mathbb R^n$ is homotopy equivalent to a point. –  Apr 14 '17 at 21:24
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    On the other hand, homotopy is already a pretty good invariant. In fact, Poincaré himself was thinking that if two closed manifolds have the same homotopy groups then they are homeomorphic ! –  Apr 14 '17 at 21:25
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Another property which is not preserved is metrizability. Any real topological vector space is contractible but not all are metrizable.

levap
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8

Few more example:

  • Local connectivity.

  • Hausdorff.

  • Compact.

  • Second countable.

Moishe Kohan
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Another topological properties not preserved under homotopy are the separation properties $T_0$, $T_1$ and $T_2$ (more known as Hausdorff).

From levap's answer, it follows that $T_0$, $T_1$ and $T_2$ are not preserved as there are real topological spaces which are neither $T_0$, $T_1$ or $T_2$.

Josué Tonelli-Cueto
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