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Suppose $X\subseteq\mathbb{R}^n$ is a connected open bounded $n$-dimensional submanifold.

1) I wonder if for the class of such spaces there is any upper bound on the LS-category of $X$?

2) Is it known that such a number would be finite, or `countable infinite'?

3) I thought if by adding the boundary points of $X$ to it, we obtain a compact space, then LS-category of $X$ is finite. But, I do not see a proof, although it might be an elementary one.

I am very much new to the LS-category business, and I would appreciate any advise on this, or pointing at any reference or an overview of the subject.

EDIT One special feature that I forgot to add is that I think of $\mathbb{R}^n$ with its Euclidean metric and like the cover to be a cover of convex sets. This is related to a similar question that I have asked a while ago Covering a space by cones

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    $\begingroup$ Could you clarify what you mean by "cover of convex sets"? For instance, an open set of $X$ is of the form $X\cap U$ for $U\subseteq\mathbb{R}^n$, and you might want either $U$ to be convex or $X\cap U$ to be convex. In either case, I'm not sure this has much to do with LS-category. $\endgroup$
    – Mark Grant
    Commented Jan 14, 2018 at 11:37
  • $\begingroup$ @MarkGrant Given a space $X$, I wanted to find (a bound for) the smallest number of convex subsets $C_\alpha\subseteq X$ so that $X=\big cup C_\alpha$. I thought if I take bounded open cones as basis for my topology on $\mathbb{R}^n$, which is equivalent to the standard one, then the LS-category could give a lower bound for this number. But, I know see that this fails, unless I can find some very unusual metric?!? $\endgroup$
    – user51223
    Commented Jan 14, 2018 at 19:20

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The LS-category of a connected, second countable, n-dimensional manifold satisfies $$cat(X)\le n+1.$$ I suppose this is well known, you find a proof in http://math.ucr.edu/~res/math246A/cuplength.pdf

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  • $\begingroup$ Thanks for the answer. One special feature that I forgot to add is that I think of $\mathbb{R}^n$ with its Euclidean metric and like the cover to be a cover of convex sets. This is related to a similar question that I have asked a while ago Covering a space by cones. $\endgroup$
    – user51223
    Commented Jan 14, 2018 at 8:31

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