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.
94 questions
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Connectedness of intersections of sets
Let $K$ be a compact subset of $\mathcal{R}^n$ with nonempty interior. Let $x_0\in \partial K$ (boundary of $K$). I want to show there exists a $\delta$ such that for any $\delta'<\delta$, $B_{\...
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Are path-connected LOTS also locally path-connected?
According to https://topology.pi-base.org/theorems/T000522 (which cites Theorem 24.1 of https://zbmath.org/0951.54001 ) every connected linearly ordered topological space (LOTS) is also locally ...
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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 $...
user410185
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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 ...
- 11.8k
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Is arbitrary intersection of locally connected topologies locally connected?
Let $X$ be a set and $\{\mathcal{T}_i \subset \mathcal{P}(X) : i \in I\}$ be an arbitrary set of locally connected topologies in $X$; it may or may not contain all of them and $I$ may be finite or ...
- 2,693
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Is intersection of refining locally connected topologies locally connected?
Let $(X, \mathcal{T})$ be a topological space, $S(X)$ be the set of topologies in $X$, and
$$\mathcal{T}^P = \bigcap \{\mathcal{T}^* \in S(X) : \mathcal{T} \subset \mathcal{T}^* \textrm{ and } (X, \...
- 2,693
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A weakly locally path connected space that is not locally path connected
I am reading through an article by Shelah where it has a definition for a weakly locally path connected space:
Say that $X$ is weakly locally path connected (WLPC) if for every $x\in X$ and every ...
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Restriction of Covering Spaces to a component.
Let $p:X\rightarrow Y$ be a covering space (i.e. locally over $Y$, $p$ looks like the projection $U\times F\rightarrow U$, with $F$ discrete).
If $Y$ is locally connected, it is known that the ...
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Connected set which is no-where path connected
Background: It's a fun exercise to try to construct a connected space $T$ such that no two points in $T$ can be connected with a path.
My solution to the puzzle was to use an order topology on a ...
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Counterexample of a disconnected topological space with open components but not locally connected
Theorem 25.3 in Munkres proved that a space is locally connected if the components of every open subset is open.
What obviously follows from the theorem is that the components of a locally connected ...
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Intuitively, what is the difference between a "simply connected" set and a "locally connected" set?
I was looking into the Mandelbrot Set and saw a note that said it has been proven that the Mandelbrot Set is "simply connected" but it is still an open question of whether or not it is "...
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Characterizing locally connected continua
I am looking for a reference to the proof of the following result:
Let $E\subseteq\mathbb C$ be a continuum. Then the following assertions are equivalent:
(1) $E$ is locally connected;
(2) (i) each ...
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Show that the n-sphere $\mathbb S^{n}$ is locally connected
A topological space $(X,\tau)$ is locally connected iff $\forall x\in X$ ,$\exists \{v_{i} : i\in I\}$ a familly of connected neighberhoods of $x$ such that $\forall u$ neighberhood of $x$ $\exists i\...
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Local connectedness is a topological property [duplicate]
I am trying to prove that local connectedness is a topological property. Then what do I need to show? Do I need to show that if $X$ and $Y$ are homeomorphic topological spaces (let $f$ be the ...
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Locally connected for sub space topology of $\Bbb R^2$
Consider the following subspace $X:=\Bbb R^2\setminus(\Bbb Q\times \Bbb Q)$ of $\Bbb R^2$, where $\Bbb R^2$ with the usual topology. I would like to check this space in the terms of various kinds of ...
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Proving/disproving a claim on the relationship between the kinds of connectedness (the usual, and path and local)
Context: I am given, as an exercise, the task of proving this claim:
For $X$ a topological space, show it is path-connected if and only if $X$ is connected and each $x \in X$ lies in some path-...
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A compact set $X$ is only locally conneted in on point but not locally connected in other points in $X$
When study the theorem of boundary correspondence in complex analysis, I come into connact with the concept of Locally Connectedness and several examples.
Definition: A compact set $X\subset\mathbb{C}$...
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topological properties of $X= \left( \cup_{n=1}^{\infty} \{\frac{1}{n}\} \times [0,1] \right) \cup (\{0\} \times [0,1]) \cup ( [0,1] \times \{0\})$
Let $X= \left( \cup_{n=1}^{\infty} \{\frac{1}{n}\} \times [0,1] \right) \cup (\{0\} \times [0,1]) \cup ( [0,1] \times \{0\}) \subset \mathbb{R}^2$.
I want to show whether this space $X$ is normal and ...
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Example of a map of coverings which is not a covering map, if the base space is not locally path-connected
Consider two coverings $X \to Y$ and $X^\prime \to Y$. A surjective map $X \to X^\prime$, which fits in a commutative triangle with the covering maps, is again a covering, given that $Y$ is locally ...
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Locally connectedness preserved after removing one point
Given a locally connected space $X$ with two or more points is it always true that for any point $x$ in $X$ the subspace $X \setminus \{x\}$ is also locally connected?
I have proved it for Hausdorff ...
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Types of connectedness for $L_n:=\{(x,\frac{x}{n})\mid0\leq x\leq1\}\subset\mathbb{R}^2$ for $n\in\mathbb{Z}_+$
Exercise: Let $L_n:=\{(x,\frac{x}{n})\mid0\leq x\leq1\}\subset\mathbb{R}^2$ for $n\in\mathbb{Z}_+$. Then consider $X=\{(1,0)\}\cup\bigcup_{n=1}^\infty L_n\subset\mathbb{R}^2$ together with the ...
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What is this topological property? A form of local connectedness?
Let $S$ be a subset of some topological space $X$.
What is known about the following property of $S$? Does it have a name? Is it standard?
Property: For all $x\in \bar{S}$ (the closure of $S$), and ...
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A sheaf is locally constant iff the etale space is a covering
Let $X$ be a locally connected space a sheaf $F$ on $X$ is called locally constant whenever each $x\in X$ has a basis of open neighbourhoods$N_x$ and if $U,V \in N_x$ such that $U$ is a subset of $V$ ...
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Is every open and connected set in $\mathbb C$ the continuous image of the open unit disk?
Let
$$
\mathbb D=\{z\in\mathbb C\ |\ |z|<1\}
$$
be the open unit disk in $\mathbb C$. It is well known that an open (nonempty) set $U\subseteq\mathbb C$ is simply connected if and only if it is ...
- 8,557
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Is the local connectedness heritable over the connected subspaces?
Definition
A topological space $X$ is locally connected if each point $x\in X$ has a base of connected neighborhoods.
So let be $X$ a locally connected space and let be $Y$ a connected subspace. So if ...
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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&...
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X has finitely many connected components, can we get X locally connected?
say X has finitely many connected components, can we get X locally connected?
I'm thinking to prove this and then I can have every component is open.
Thanks!
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Proof Verification : Equivalent Condition for Locally Connected Space
A topological space $X$ is locally connected if
for every $x$ in $X$ and for every open set $V$ containing $x$, there is a connected open set $U$ with $x \in U \subset V$.
I think it is equivalent ...
- 1,203
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Local connectedness of topologist's comb
I'm using Croom's textbook, and came with the following definition of local connectedness:
A space $X$ is locally connected at a point $p$ in $X$ if every open set containing $p$ contains a connected ...
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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
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Show that if $ X $ has the property $ \mathcal{M} $, then $ X $ is locally connected.
Let $ (X, \tau) $ be a topological space. We will say that $ X $ has the property $ \mathcal{M} $ if every open coverage of $ X $ admits a finite refinement consisting of connected sets.
Show that if $...
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If every open cover of $ X $ admits a finite refinement consisting of connected sets, then $ X $ is locally connected.
Let $ (X, \tau) $ be a topological space. If every open coverage of $ X $ admits a finite refinement consisting of connected sets, then $ X $ is locally connected.
Proof:
let $\{U_i\} $a open coverage ...
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$X$ a compact Hausdorff space has the property $ \mathcal{M} $ if, and only if $X$ is locally connected.
$(X,\tau)$ a topological space. $X$ a compact Hausdorff space has the property $ \mathcal{M} $ if, and only if $X$ is locally connected.
Having the $ \mathcal{M} $ property means:
Let $ (X, \tau) $ ...
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X is connected and locally connected
Let $X$ be a $T_1$ topological space. Show that $X$ is connected and locally connected if and only if for each open cover $\{U_s\}_{s\in S}$ and any $x, y \in X$, there exist $s_1,. . . , s_n \in S$ ...
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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 ...
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60
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the boundary of a component $C$ of $A$ is contained in the boundary of $A $.
Let $(Y, \tau)$ be a locally connected topological space and $A\subset Y$. Let $C$ be a component of
$A$. Show that $\partial C\subset \partial A$.
I proved that $Int_{Y}(C)= C\cap Int_{Y}(A)$. Using ...
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$ X = A \cup B $ where $ A $ and $ B $ are closed and $ A \cap B $ is locally connected. Show that $A$ and $B$ are locally connected.
Let $(Y, \tau)$ a locally connected topological space. suppose $ Y = A \cup B $ where $ A $ and $ B $ are closed and $ A \cap B $ is locally connected. Show that $A$ and $B$ are locally connected.
...
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