Let $X$ be a topological space, let $\mathrm{Map}(X)$ be the set of all continuous maps $X\to X$, equipped with the compact-open topology, and finally, let $\mathrm{Homeo}(X)\subset \mathrm{Map}(X)$ be the subset of homeomorphisms.
Even for very nice spaces it seems like $\mathrm{Homeo}(X)$ may fail to be an open subset of $\mathrm{Map}(X)$. For example on $X=\mathbb R$ we may approximate the homeomorphism $x\mapsto x^3$ with the non-injective maps $x\mapsto (x^2-\epsilon^2)x$. We can transfer this example to $\mathbb S^1$ via stereographic projection and further use that composition in $\mathrm{Homeo}(\mathbb S^1)$ is continuous to show that:
$\mathrm{Homeo}(\mathbb S^1)$ has empty interior inside $\mathrm{Map}(\mathbb S^1)$
This is in stark contrast to the differentiable world, where spaces of diffeomorphisms (say on a compact manifold) are open in the ambient spaces of $C^1$-maps.
I am curious what else is known about the topology of $\mathrm{ Homeo}(X)$, viewed as a subset of $\mathrm{Map}(X)$. Is the highlighted statement true in general (for some class of spaces), can it be strengthened (e.g. is $\mathrm{Homeo}(X)$ nowhere dense?) or are there other interesting phenomona that highlight the difference to the differentiable world?