Q:

Homeomorphic vs Homotopy Equivalent

Now I know the definition of each of these, and I also know that homeomorphic implies homotopy equivalent.
However I seem to be coming across examples of spaces that are homeomorphic but not homotopy equivalent.
For example two interlocked circles are homeomorphic to two disjoint circles, via an obvious map, but you can't continuously deform the interlocked circles into the disjoint ones, so doesn't that mean they aren't homotopy equivalent?
I don't think I fully understand homotopy equivalence.

A:

You're mixing up homotopy equivalence of spaces with homotopy equivalence of maps. What you're talking about are two maps $S^1\to X$ where $X$ is $\Bbb R^3$ with a circle removed. Those maps are not homotopy equivalent; there is certainly no continuous way to deform an $S^1$ that is interlocked with the removed circle into one that isn't.
However, the space itself, i.e. the disjoint union of two circles without any reference to an ambient space is both homeomorphic and homotopy equivalent to itself.

