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By "almost" I mean that it can be slightly perturbed within a given fixed compact domain to obtain a homeomorphism from that domain to its image.

To be more precise:

Let a compact topological space $U \subset \mathbb{R}^n$, and let a map $f:U\to M$ such that for any $\epsilon>0$ there exists homeomorphism $f_\epsilon:U\to V$ (continuous function with continuous inverse) between $U$ and $V$ for some $V\subset \mathbb{R}^n$ such that $$ \|f(x)-f_\epsilon(x)\|_2<\epsilon \text{ for all } x \in U. \tag{1} $$

Examples:

  • a constant map from cube $x \in [0,1]^3$ to $\mathbb{R}^3$, $f(x): x \mapsto (0,0,0)$ is almost homeomorphism on the cube. Indeed, for any $\epsilon$ one can introduce $f_\epsilon(x): x \mapsto \frac{1}{2\sqrt{3}} \epsilon( x_1, x_2, x_3)$ which is a homeomorphism, and (1) is satisfied.

  • a function from $\mathbb{R}\to \mathbb{R}$, $f(x):x^2$ is not almost homeomorphism on $[-1, 1]$, since there is no inverse map for $x=0$.

In one dimension, I presume, a function is almost homeomorphism if and only if it is monotonic, but I'm not familiar with the notion of monotonic in higher dimensions.

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  • $\begingroup$ Monotonic, not monotonous! $\endgroup$
    – KCd
    Apr 25 '19 at 18:48
  • $\begingroup$ @KCd thanks, fixed. $\endgroup$
    – them
    Apr 25 '19 at 19:18
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The term is "near homeomorphism".

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