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Results tagged with gt.geometric-topology
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user 53155
Topology of cell complexes and manifolds, classification of manifolds (e.g. smoothing, surgery), low dimensional topology (e.g. knot theory, invariants of 4-manifolds), embedding theory, combinatorial and PL topology, geometric group theory, infinite dimensional topology (e.g. Hilbert cube manifolds, theory of retracts).
1
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Is it possible to connect every compact set?
This is meant to fill in some of the details outlined by Anton Petrunin's answer, and also to refine the statement slightly. Recall that a compact connected Hausdorff space is called a continuum.
We …
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0
votes
Is it possible to connect every compact set?
Here is another answer, based again on Anton Petrunin's idea, but obtaining a slightly different result.
Theorem. Let $X$ be a connected and locally path-connected completely metrizable space. Then f …
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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 …
- 5,529
3
votes
1
answer
142
views
Two paths to the boundary with no holes in between
Let $X\subset \mathbb{R}^2$ be open connected (and let's say bounded), let $x\in X$ and $y\in\partial X$ be such that there is a Jordan curve $\gamma:[0,1]\to X\cup\{y\}$ such that $\gamma(0)=x$ and $ …
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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 …
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4
votes
0
answers
226
views
Enlarging a compact set in order to improve its shape
In my previous question it was established that if $X$ is a metrizable, connected, locally path connected space and $K\subset X$ is compact, then there is a Peano continuum $L\subset X$ such that $K\s …
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7
votes
1
answer
228
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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 …
- 5,529
2
votes
1
answer
131
views
Approximate Jordan-Brouwer theorem
This came up when thinking about this question.
It is well-known that the image of a homeomorphic embedding $f:S^n\to \mathbb{R}^{n+1}$ separates the space into exactly two components, one of which is …
- 5,529
1
vote
1
answer
102
views
Approximate Jordan-Brouwer theorem (corrected)
My first attempt to ask this question sort of failed (I'll explain below).
This came up when thinking about this question.
It is well-known that the image of a homeomorphic embedding $f:S^n\to \mathbb …
- 5,529
3
votes
0
answers
72
views
A holomorphic shrinking of a domain into a compact subset
This question is related to these two.
Let $X\subset \mathbb{C}^{n}$ be a bounded domain. I am interested in the following property: there is a sequence of continuous maps $\varphi_n:\overline{X}\to X …
- 5,529
1
vote
0
answers
53
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Spaces that are comparable with their compacts
This is an outgrowth of this question.
For a (metrizable) space $X$ consider the following increasingly strong properties:
(i) For every compact $K\subset X$ there is a map $f:X\to X$ such that $K\s …
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2
votes
0
answers
82
views
Enveloping a Jordan curve with a trace of another one
This question is inspired by this one, or rather the way I understood it.
Let $\gamma$ and $\delta$ be parametrised Jordan curves on the plane (i.e. homeomorphisms from $S^1$ onto its image in $\mat …
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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 …
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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). …
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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 …
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