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Results tagged with gn.general-topology
Search options questions only
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user 53155
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 …
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 …
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 …
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 …
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 …
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 …
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). …
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 …
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 …
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 …
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 …
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 …
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 …
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 …
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 …