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I think the following is true and I need a reference for the proof. (Given a closed surface $S$, i.e. a compact 2-dimensional topological manifold (without boundary), we endow $S$ with a distance generating its topology, and endow the set of self-homeomorphisms of $S$ with the distance max(uniform distance between two maps, uniform distance between their inverses)).

For any $\varepsilon>0$ there exists $\eta>0$ such that for any self-homeomorphism $f$ of $S$ whose distance to the identity is $<\eta$, there is an isotopy $t\mapsto f_t$ from $f$ to the identity that stays at distance $<\varepsilon$ to the identity.

I'd be interested in a reference and/or a hint of the proof.

Note: Unless I'm too tired and got it wrong, this result implies (and is implied by the fact) that the group of self-homeomorphisms of $S$ is locally path-connected (in the sense that every point has a path-connected neighbourhood, not necessarily open), whence the title of this question.

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This is a particular case of Corollary 1.1 of Edwards, Robert D.; Kirby, Robion C. Deformations of spaces of imbeddings. Ann. of Math. (2) 93 (1971), 63--88. MR0283802, which says that the group of homeomorphisms of any compact manifold is locally contractible.

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In the particular case of surfaces, I found the following reference which includes a proof that is not too complicated: Regular Mappings and the Space of Homeomorphisms on 2-Manifolds by Hamstrom and Dyer. They prove local contractibility, which is more than I asked. It works for surfaces with or without boundaries and includes a slight generalization with fixed points on the boundary. This is Theorem 1 in this article. The proof fits in 6 pages, is a bit heavy on notations but this remains managable. Unfortunately there is no figure. The proof uses conformal maps for a couple of lemmas, Alexander's trick, and a technique due to J.H. Roberts (Local arcwise connectivity in the space $H^n$ of homeomorphisms of $S^n$ onto itself, Summary of Lectures, Summer Institute on Set Theoretic Topology, Madison, Wisconsin, 1955, p. 100) but I cannot find the corresponding reference. They also cite a German article of Kneser (Die Deformationssätze der einfach zusammenhägenden Flächen, Mathematische Zeitschrift, Vol. 25(1926), pp. 362-372) as a source of inspiration, but my knowledge of German is very basic so reading it would represent quite an investment.

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  • $\begingroup$ Link to Hamstrom–Dyer (1958) behind paywall: projecteuclid.org/euclid.dmj/1077468064 $\endgroup$ – YCor Oct 15 '20 at 8:42
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    $\begingroup$ Surprising given it was published 62 years ago. I hope Duke Math J will follow an Open Access model soon. $\endgroup$ – Arnaud Chéritat Oct 15 '20 at 9:29

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