All Questions
Tagged with locally-connected compactness
10 questions
1
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Number of components of an open subspace of a compact locally connected space
In a locally connected space $X$, one can show that the connected components of any open subspace $U\subseteq X$ are all open in $X$ (cf. Theorem 25.3 in Munkres' Topology 2e). Therefore, if $X$ is ...
- 1,269
4
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1
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171
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Metric space that can be written as the finite union of connected subsets but isn't locally connected
I'm looking for an example of a metric space $X$ such that for every $\epsilon > 0$ there exist connected subsets $A_1, \dots A_n$ for some $n \in \mathbb{N}$ such that $X = \cup_{i = 1}^nA_i$ and ...
- 891
14
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2
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235
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Does locally compact, locally connected, connected metrizable space admit a metric with connected balls?
If $X$ is a locally compact, locally connected, connected metrizable space, does that imply that there must be a metric $d$ on $X$ such that $B(x, r) = \{y\in X : d(x, y) < r\}$ is connected for ...
- 11.9k
2
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1
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100
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Boundaries in Spaces where Quasicomponents and Components Coincide
Let's call a space $X$ geometric if its components and quasi-components coincide. Let's also define a property called the boundary bumping property:
$X$ has the boundary bumping property ("bbp&...
- 383
2
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0
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57
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Metric space locally compact but not uniformly locally connected
Definition: A set $M$, also in a metric space, is said to be uniformly locally
connected if and only if for every $\varepsilon > 0$ there exists $\delta>0$ such that any
pair of points $x, y$ of ...
- 143
5
votes
1
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98
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$X$ is locally connected and countably compact
Let $(X,\tau)$ be a topological space $T_3$. Show that the following statements are equivalent:
Every open and finite coverage of X has a finite refinement consisting of connected sets.
Space X is ...
- 504
1
vote
1
answer
89
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Maps between Peano spaces
If $X$ is a Peano spaces with more than one point, and $Y$ is any Peano space, there exists a continuous map $f$ from $X$ onto $Y$.
Also, if $a,b$ are distinct points in $X$ and $c,d$ are distinct ...
- 3,644
1
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1
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88
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Hausdorff and locally compact
Theorem A space $X$ is locally compact and Hausdorff if and only if it is
homeomorphic to an open subset of a compact Hausdorff space.
Can any one give me hint to prove this result. I want the ...
user787636
4
votes
1
answer
589
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Characteristic of a connected, locally compact, Hausdorff space X which is locally connected.
Q. Prove that a connected, locally compact, Hausdorff space X is locally connected if and only if for each compact subset K and each open set U containing K, all but a finite number of components of X-...
- 41
1
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2
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130
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Set of Points where X Fails to be Locally Connected
I am stuck on a problem! Suppose $X$ is a compact, connected metric space. Let $L(X)$ be the set of points at which $X$ is not locally connected (here, locally connected means the point has a local ...
- 383