Questions tagged [connectedness]

Use this tag for question on connected spaces and various related notions (connected components, locally connected spaces, pathwise/arcwise connected spaces, totally disconnected spaces ...) and for connectedness in graph theory.

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Connected Metric Spaces: Strategies

I am not really sure if my ideas in this topic are correct. Can anyone help me? Finding the connected components of a metric space $X$. Suppose there are two connected components $C_1, C_2$ of $X$. ...
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$H:M\rightarrow \mathbb{R}$ is a continuous map from top space $M$. Show that $H^{-1}(e)$ divides $M$ where $e\in \text{int}(H(M))$

Let $M$ be a topological space and $H:M\rightarrow \mathbb{R}$ be continuous and surjective. Suppose $e\in \text{int}(H(M))$. Then show $H^{-1}(e)$ divides $M$; that is, $M\setminus H^{-1}(e)$ has ...
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How does this set equality follow?

This is proposition 4.9 in Lee's intro to topological manifolds. What I don't understand is the proof of part b where it is said that $$X = \overline{A} \subseteq{\overline{U}} = U$$ In particular, ...
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Non local connected $\implies$ Non-open component example [closed]

I'm trying to figure out various definitions of local connectedness, here, proved equivalence of definitions of locally connected space. A topological space $T=(𝑆,\tau)$ is locally connected if and ...
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Showing particular quotient space is (not) compact, connected and Hausdorff

I came across the following question, I'm unsure about some of my answers. Let $U = \{(x, y) \in \mathbb{R}^{2} \mid y \in \{0, 1\}\}$ be a subspace of $\mathbb{R}^{2}$. We define the following ...
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Are these Cayley graphs strongly connected?

My Background: I have no formal training in graph theory. I'm just a group theory PhD student. The Story: I attended the New Perspectives in Computational Group Theory conference last week in honour ...
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Differential Forms and Path Integrals

How to prove this problem from Fulton Book (Algebraic Topology) Show that an open set U in the plane is connected if and only if there is a segmented path between any two pints of U. Can you show that ...
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Proving that Intervals are connected in $\Bbb R$ using continuous functions into $\{0,1\}$

I have come up with a proof for the fact that Intervals are connected sets in $\Bbb R$, that utilizes an equivalent definition of connected sets. Namely: Definition. A topological space $X$ is called ...
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Connect two points on an annulus

I want to prove that the annulus given by the set $$B(a;R_1,R_2) =\{{z \in \mathbb{C} : R_1 < |z-a| < R_2\}}$$ is a connected space. As I am working in $\mathbb{C}$ it is enough to prove it to ...
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Is Rudin's definition of simple connectedness in his RCA equivalent to the usual one?

At the bullet point $10.38$ in his Real and Complex Analysis, Rudin gives a very quick rundown on homotopy and simple connectedness, but the definition he gives is the following: If [a topological ...
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Is there a classification of connected sets in $\mathbb{R}^2$? [duplicate]

We know that in $\mathbb{R}$ with standard topology, a subset $A$ of $\mathbb{R}$ is connected if and only if $A$ is an interval (including points, rays, $\mathbb{R}$). Is there a classification of ...
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Numner of Graph functions

Let, T(V, E) be an undirected tree with n vertices, where V={$v_1,v_2,...,v_n$}. Let d(i)represent the degree of verticex $v_i$. Let f be a function from V to {1,2,...,n}. Compute the number of such ...
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There exist a path of length $\chi(G)- 1$ in a connected graph G

For any undirected connected graph $G$, let $\chi (G)$ be its chromatic number. Then for every vertex $v$ in $G$, there exists a path of length $\chi (G) - 1$ starting from $v$. My approach : for ...
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A countable metric space implies totally disconnectedness

I'm stuck on a problem in general topology. But first, let's establish some observations: Let $(X, \tau)$ be a topological space. The connected component of $x \in X$, denoted by $C(x)$, is the union ...
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Given the sets $X:(x+1)^2 + y^2 \leq 4, x^2+y^2 \ge 1$ and $Y:1 \leq x^2 + y^2 \leq 4$. Is X homeomorphic to Y?

I'm trying to prove they aren't via connectedness. I figured the point (1,0) is an important point (cut point) in this problem, but i think that X without (1,0) is still connected.
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Is $\mathbb R/\mathbb N$ homeomorphic to $\mathbb R/\mathbb Z?$

Where $\mathbb R/\mathbb N$ is the quotient topological space, where $\mathbb N$ is collapsed onto a point. Same with $\mathbb R/\mathbb Z.$ I was thinking I could prove it wasn't homeomorphic using ...
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Can the Euclidean unit interval have a finer connected topology?

Consider the unit interval $X = [0,1]$ equipped with the Euclidean topology $\tau_E$, and consider some other topology $\tau_F$ on $X$ that is strictly finer than $\tau_E$. Does there exist such a ...
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What can't be added for "maximally connected graph" to keep some property said in definition?

In wikipedia it says A graph is said to be maximally connected if its connectivity equals its minimum degree. A graph is said to be maximally edge-connected if its edge-connectivity equals its ...
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Accessibility Components does *not* coincide with Path Connected and Connected Components for Length Spaces

I am working through "A Course in Metric Geometry" and I do not believe this exercise to be true: Exercise $\boldsymbol{2.1.3}$ $3)$ Verify that accessibility components coincide with both ...
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Is there a nondiscrete linearly ordered topological space (LOTS) that is extremally disconnected?

A space is extremally disconnected if the closure of every open set is open. I was wondering if there is a ordered space that is extremally disconnected but not discrete. Currently there is no such ...
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Path existence between local maxima

Let $f:\mathbb{S}^2\to\mathbb{R}$ be a smooth function and let $M=\{x_1,\dots x_K\}$ be the set of its local maxima. I am trying to determine if there is a way to find out if there exists a '...
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