Topological properties that aren't conserved over homotopy?

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?

Solution 1:

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

Solution 2:

Another property which is not preserved is metrizability. Any real topological vector space is contractible but not all are metrizable.

Solution 3:

Few more example:

• Local connectivity.

• Hausdorff.

• Compact.

• Second countable.

Solution 4:

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 $T_2$.