# “monotone” homotopy?

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:

1. Has anybody seen such "monotone homotopies" somewhere in the literature? What is the terminology for them?
2. 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.

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Interesting question. It took me a little time to find the infinitely many classes for $I$ with two points! What does the word "monotone" refer to? –  Jim Conant May 13 '13 at 13:41
I have not seen this before, but you may be interested in looking up "controlled homotopy theory". –  Vidit Nanda May 13 '13 at 14:09
@ Jim Conant, "monotone" is a purely temporary technical term, coming from the feeling that the homotopy runs in some 'monotone', in fact continuous manner over the set A. There should be some more suitable term though. I suppose that maore interesting examples exist in higher dimensions. –  reader2 May 13 '13 at 15:07
@Vidit Nanda, thank you for the clue, I'll follow up –  reader2 May 13 '13 at 15:20
My instant reflex would be to consider the continuity of $t\mapsto f^{-1}_t(A)$ in the space of closed sets (with the Hausdorff metric) rather than just continuity of the distance for the purpose of the definition of the monotone homotopy. –  Włodzimierz Holsztyński May 14 '13 at 1:23