# In topology: Is there a special terminology for a map that is "almost'' a homeomorphism?

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.

• Monotonic, not monotonous!
– KCd
Apr 25 '19 at 18:48
• @KCd thanks, fixed.
– them
Apr 25 '19 at 19:18