This is a question about a concept that I call "monotone homotopy" which arises in a natural way in some topological situations.

Let $X$ be a (bounded) metric space, $Y$ be a topological space and $A\subset Y$ be its closed subset. We say that two maps $f_{0},f_{1}:X \to Y$ are "monotonically homotopic" over $A$ if there is a homotopy between them $f_{t}:X \to Y$, $t\in [0,1]$, such that the set $f_{t}^{-1}(A)$ is moving continuously in the Hausdorff distance in $X$, that is, the function $\varphi(t)=d_{H}(f_{t}^{-1}(A),f_{0}^{-1}(A))$ is continuous in $t$. Here, $d_{H}$ is the Hausdorff distance in $X$ induced by the metric.

This concept is, of course, a refinement of the usual homotopy classes. It turns out that even in very simple situations, a lot of "monotone" homotopy classes may arise (even if $Y$ is contractible). For example, if $X=Y=[0,1]$ and $A$ is a 2-element subset of $Y$, there are infinitely many such classes. Another example: if $X=Y=\mathbb{S}^{1}$ and $A=\{p_{0}\}$ is a point, then in any usual homotopy class of maps $\mathbb{S}^{1} \to \mathbb{S}^{1}$ there are infinitely many "monotone" classes over $A$.

Similarly, one may consider monotone homotopies over a system $\{A_{\alpha}\}$ of subsets of $Y$ and this is in fact the situation I am interested in.

**Questions:**

- Has anybody seen such "monotone homotopies" somewhere in the literature? What is the terminology for them?
- Do you think that this concept has any applications? Is it sufficiently interesting to warrant further investigations?

Addendum: I suppose that $d_{H}(M,\varnothing)=\infty$ for a nonempty set $M$ and $d_{H}(\varnothing,\varnothing)=0$, which is important for the definition. So, two maps $f,g:X \to Y$ such that $f^{-1}(A)=\varnothing$, $g^{-1}(A)\neq\varnothing$ cannot lie in the same monotone class.