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Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.

14 votes
1 answer
444 views

Does existence of midpoints imply intrinsic?

It is well-known, that a complete metric space, where any two points have a midpoints ($\forall x,y~ \exists z:~d(x,z)=d(y,z)=\frac{d(x,y)}{2}$) is strictly intrinsic, in the sense that any $x,y$ can …
erz's user avatar
  • 5,529
13 votes
2 answers
2k views

When can we divide continuous functions?

Let $X$ be a compact Hausdorff topological space such that for every continuous $f,g:X\to\mathbb{R}$ with $0\le f\le g$ there is a continuous $h:X\to\mathbb{R}$ such that $f=gh$. What can be said abo …
erz's user avatar
  • 5,529
9 votes
2 answers
750 views

Is limit of null-homotopic maps null-homotopic?

The question is motivated by my failed comment to this one. Let $M$ and $N$ be path connected locally compact, locally contractible metric spaces (you may assume that they are manifolds). Let $\varphi …
erz's user avatar
  • 5,529
7 votes
2 answers
611 views

Which topological spaces contain dense simply connected subspace?

And when can this subspace be chosen to be open? As the answer to this question indicates, any manifold contains an open dense subset, which is homeomorphic to $\mathbb{R}^{n}$, and so for manifolds …
erz's user avatar
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7 votes
1 answer
228 views

Retracting off a compact set

Let $K$ be a compact set in $\mathbb{R}^n$ and let $U$ be a bounded open set that contains $K$. You may assume both are connected. Can we always find an open $V$ such that $K\subset V\subset\overl …
erz's user avatar
  • 5,529
6 votes
1 answer
554 views

Is restriction a closed map?

Originally asked on MSE. Let $X$ be a normal (or even metrizable) topological space and let $Y$ be a closed subset of $X$. Let $C(X)$ be the linear space of all continuous scalar functions on $X$ end …
erz's user avatar
  • 5,529
6 votes
1 answer
487 views

Map which is null-homotopic on compacts

This is the missing ingredient towards answering my previous question. Let $M$ and $N$ be path connected locally compact, locally contractible metric spaces (you may assume that they are manifolds). …
erz's user avatar
  • 5,529
6 votes
1 answer
244 views

Is the projectivization of a topological vector space Tychonoff?

Let $E$ be a locally convex topological vector space over $\mathbb{R}$. The projectivization $PE$ is the quotient of $E\backslash\{0_{E}\}$ with respect to the equivalence relation $e\sim f$ if $e=\la …
erz's user avatar
  • 5,529
5 votes
1 answer
255 views

Generating the topology of a manifold

Let $X$ be a topological manifold of dimension $d$, and let $F$ be a collection of continuous maps from $X$ into $\mathbf{R}^d$ such that: $F$ separates points of $X$, i.e. for any two distinct poin …
erz's user avatar
  • 5,529
5 votes
0 answers
144 views

Do products of distance functions separate points?

Let $(X,d)$ be a metric space without isolated points and of diameter $1$. Let $Y=\{y_m\}_{m=1}^{\infty}$ be a dense subset of $X$. Define $g_0\equiv 1$, and for $m>0$ let $g_m=d(\cdot,y_1)\dotsm d(\c …
erz's user avatar
  • 5,529
5 votes
1 answer
205 views

If a subspace $F$ is contained in a subspace $G$, and $H$ is close to $G$, can we choose a s...

Let $E$ be a Banach space. Recall that the collection of all closed linear subspaces of $E$ can be turned into a metric space in a number of ways. In particular, consider the notion of a gap: if $G$ a …
erz's user avatar
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5 votes
1 answer
313 views

Does every open set contain a dense $F_{\sigma}$ subset?

Let $U$ be a regular open set in a Tychonoff space $X$ (regular means that it is an interior of a closed set). [ In my specific situation $U$ is of the form $\operatorname{int} f^{-1}(0)$, where $f$ i …
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5 votes
2 answers
438 views

Space of curves

I am reading Burago, Burago & Ivanov's book where they distinguish the notion of a curve and a path in the following way: a path in a topological space $X$ is simply a (continuous) map from a connecte …
erz's user avatar
  • 5,529
5 votes
3 answers
699 views

Is it possible to connect every compact set?

Let $X$ be a "nice" space: metrizable, connected, locally path connected perhaps. Let $K\subset X$ be a compact set. Is there a always a compact connected $L\subset X$ such that $K\subset L$? Th …
erz's user avatar
  • 5,529
4 votes
1 answer
224 views

Bounded growth of functions vs bounded growth of functions on countable sets

I am wondering if the boundedness of growth can be characterized by sequences. I am not sure if I use the term "growth" correctly, or use the correct tags for this question. Here is what I mean. Let …
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