Questions tagged [locally-connected]
For questions on locally connected topological spaces. A topological space is called locally connected if every neighborhood of every point contains a connected open neighborhood.
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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 ...
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Is an open connected subspace of $\mathbb{R^2}$ locally path-connected?
I want to proof the following statement :
Let X be an open connected subspace of $\mathbb{R^2}$. Show that X is also path connected. The standard way to prove this problem from what I at least saw was ...
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Adding point to connected open set
Let $X$ be a compact, connected, locally connected space. Let $U$ be a connected open subset of $X$. Let $p\in \overline U$. Clearly $U\cup \{p\}$ is connected.
Is $U\cup \{p\}$ locally connected?
Is $...
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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 ...
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Definition of local connectedness by Zorich's Mathematical Analysis II
I am reading chapter 9,section 4,of Mathematical Analysis II, written by Zorich.
In Exercise 4 of the fourth section he defined the locally connectedness: a topological space $\left(X,\tau\right)$ is ...
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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 ...
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Proof about locally connected
I have a doubt with my proof of the following result:
A topological space $(X,\mathcal{F})$ is locally connected if and only if for every open set $U \subset X$ each connected component of $U$ is open ...
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Prove that product of two locally connected spaces is locally connected. [duplicate]
Let $X$ and $Y$ be two locally connected spaces. I need to show that $Z = X \times Y$ is locally connected. Here is my attempt:
Proof. Let $z=(x, y)\in Z$ and $N$ be any neighborhood of $z$. I need to ...
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Is pointwise locally connectedness preserved under a quotient map?
Let $(X, \mathcal{T}_X)$ and $(Y, \mathcal{T}_Y)$ be topological spaces. $X$ is locally connected at $x \in X$, if for each $U \in \mathcal{T}_X(x)$ there exists $U' \in \mathcal{T}_X(x)$ such that $U'...
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Quotient of R by irrationals with same absolute value
while studying some general topology I came across the following topological space: on $\mathbb{R}$ we define an equivalence relation by $x \sim y$ if and only if $x=y$ or $|x|=|y|$ and $x \notin \...
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Potential errata found: show the non locally connected points of a metric space form a discrete space, under a given new metric
I am studying Topology and Groupoids by Ronald Brown, 3rd Edition. I believe I have spotted an errata in a given question, as I seem to have found a counterexample to what it wishes me to prove. I ...
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Consider the space $(\Bbb R, \tau_l)$, where $\tau_l$ is the lower limit topology. Is this space locally connected?
Consider the space $(\Bbb R, \tau_l)$, where $\tau_l$ is the lower limit topology. Is this space locally connected?
I first thought that this would be true since $\tau_l$ is finer than the standard ...
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Let $X$ be a locally connected space and $f:X \to Y$ a continuous closed surjection. Show that $Y$ is locally connected.
Let $X$ be a locally connected space and $f:X \to Y$ a continuous closed surjection. Show that $Y$ is locally connected.
Let $V \subset Y$ be an open set and $C$ a component containing $V$. To prove ...
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Let $X=\{(x,y) \in \Bbb R^2 \mid x = 0 \text{ or } x^2+y^2 \in \Bbb Q\} \subset \Bbb R^2$. Is $X$ connected? Is it locally-connected?
Let $X=\{(x,y) \in \Bbb R^2 \mid x = 0 \text{ or } x^2+y^2 \in \Bbb Q\} \subset \Bbb R^2$. Is $X$ connected? Is it locally-connected?
$X$ is the $y$-axis union rational points on the unit disc. I ...
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neighbourhood open basis for a connected set in $\mathbb C$
Given $X$ a connected and locally connected space, let $S\subseteq X$ a connected set.
Given any open $U$ containing $S$, is it true that there exists an open and connected $V$ such that $S\subseteq V ...
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