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

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About level curves of a continuous function in a real square, and connectivity

Assume f is a continuous function on the (unit) square in real plane. Name the edges N,S,E and W in the natural way. Assume f is >0 at W edge and <0 on E edge. Intuitively it is clear that there ...
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31 views

Connected Sets on Metric Spaces

I'm taking a first course in real analysis, and we're using Rudin's Principles of Mathematical Analysis as our main (only) book. In chapter two, Rudin discusses basic topology from the point of view ...
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31 views

X × Y \ {(x, y)} is path connected if X and Y are both path connected [on hold]

i was solving exercise questions and came across this problem on connectedness ...Let X and Y be path connected spaces and (x, y) ∈ X × Y . If each X and Y has more than one elements then X × Y \ {(x, ...
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29 views

tupules $(x,y)$ with at least one entry rational is connected in $R^2$

I have studied connectedness and came across a problem which goes like this.. all the tuples $(x,y)$ with at least one entry rational is connected in $\Bbb R^2$. I have tried to prove it by ...
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1answer
57 views

Is “connected, simply connected” Redundant?

Here are my definitions of "connected" and "simply connected." A topological space $X$ is connected if and only if it is not the union of two nonempty disjoint open sets. A topological space ...
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1answer
39 views

Two definitions of connectedness: are they equivalent?

A topological space $(X, \tau)$ is connected if $X$ is not the union of two nonempty, open, disjoint sets. A subset $Y \subseteq X$ is connected if it is connected in the subspace topology. In ...
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1answer
38 views

A topological space is extremally disconnected iff every two disjoint open sets have disjoint closures

Show that for any topological space $X$ the following are equivalent: $X$ is extremally disconnected Every two disjoint open sets in $X$ have disjoint closures. My attempt at a ...
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22 views

What is the smallest $5$-vertex-connected ($5$-edge-connected) planar graph?

A planar graph cannot be $6$-connected because the number of edges of a planar graph with $n$ vertices is at most $3n-6$, while a $6$-connected graph with $n$ vertices must have at least $3n$ edges. ...
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73 views

Show that f is onto.

Let $X$ be a compact connected Hausdorff space and $f:X\rightarrow X$ a continuous open map. Show that f is onto.
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1answer
14 views

Linear-time algorithm for deciding triconnectivity?

The german site of wikipedia (Look at wikipedia k-zusammenhang) states that there are linear-time algorithms to decide whether a given undirected graph is triconnected (Deleting any two vertices ...
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1answer
66 views

If the partial derivatives are $0$ is a function constant?

I am trying to prove that if we have a differentiable function: $f:\mathbb{R}^2\rightarrow \mathbb{R}$, and the partial derivatives of f is 0, then f is constant on a connected set. I am using the ...
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1answer
22 views

Proving graph connectedness given the minimum degree of all vertices

I know that this is a repeat of a previous question asked with a similar title, but I didn't want to revive an old thread. The solution presented in that thread seems to be the common one, but I was ...
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1answer
51 views

Prove that a graph with $n$ vertices is connected if it has $(n^2 - 3n + 4)/2$ edges.

Let $G$ be a graph with $n > 2$ vertices with $(n^2 − 3n + 4)/2$ edges. Prove that $G$ is connected. This is my assignment question and I have no idea how to approach. Please suggest something.
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41 views

Why do the numbers not coincide?

$$A095983$$ Number of biconnected labeled graphs on n nodes. $$0, 0, 1, 10, 253, 11968, 1047613$$ $0\ 0$ $1\ 0$ $2\ 1$ $3\ 10$ $4\ 253$ $5\ 11968$ $6\ 1047613$ My enumeration with free turbo ...
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18 views

Irreducible matrices and connected graphs

The adjacency matrix of a simple undirected graph is irreducible if and only if the graph is connected. Here my questions : Is there an efficient method to check whether a matrix is irreducible (I ...
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1answer
35 views

Smallest non-hamiltonian k-connected graph

Let $G$ be a simple undirected $k$-connected graph with at least $k+1$ vertices. What ist the least number of vertices, $G$ can have, if it is not hamiltonian ? I know Tutte's theorem that every ...
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1answer
40 views

What do you call a graph where all nodes are reachable?

simple question. What's the term for an (undirected) graph, where one can reach any other node from any node?
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25 views

Conditions for a $2$-connected graph to be hamiltonian?

Let G be a simple undirected graph. If G is connected, and every vertex has degree $2$, then G is hamiltonian. (In fact, G only consists of the hamilton-circle) Are there some weaker sufficient ...
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2answers
48 views

Connected topological spaces, product is connected

Show that if $(X_i)_{i \in \mathcal I}$ where $X_i$ is a topological space for every $i \in \mathcal I$, then $X_i$ is connected for every $i$ if and only if $\prod_{i \in \mathcal I} X_i$ is ...
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3answers
84 views

Openness of path connected components of open subsets of $\mathbb C$

Let $\Omega\subset \Bbb{C}$ be an open set. My textbook states that every path connected component of $\Omega$ is open. I can't seem to understand why that is. Why does every point have to contained ...
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1answer
20 views

The intersection of a connected subspace with the boundary of another subset

Can someone please verify my proof or offer suggestions for improvement? Definition/Notation: The boundary of $A$, denoted by $\operatorname{Bd}(A)$, equals $\overline{A} \cap \overline{X-A}$. ...
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2answers
40 views

Connectedness and non-local-connectedness of a subspace of $\mathbb R^2$

Let $(X,\tau)$ be the subspace of $\mathbb R^2$ consisting of the points in the line segments joining $(0,1)$ to $(0,0)$ and to all the points $(1/n,0)$, $n=1,2,3,\ldots$. Show that $(X,\tau)$ ...
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1answer
46 views

Cardinality of fibres of covering maps of connected spaces

If I have a covering map $p:E \rightarrow B$ and some connected set $U$, that is evenly covered, then $p^{-1}(U)$ as a partition into slices is unique. Now, if I assume that $B$ is connected, then I ...
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1answer
36 views

Connected subsets of metric (or T1) spaces

I have proved some statements about connected subsets of a metric space. They are really basic, but I want to make sure that they are true. Would someone please tell me whether these statements are ...
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1answer
43 views

Connectedness of both $Y \cup A$ and $Y \cup B$ where $A, B$ is a separation of $X -Y$

Let $Y\subset X$ be such that both $X$ and $Y$ are connected. Show that if $A$ and $B$ is a separation of $X-Y$, then $Y\cup A$ and $Y\cup B$ are connected. I found a proof for this problem in this ...
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1answer
42 views

Construction of the simply connected Lie group of a given Lie algebra

Given a finite dimensional real Lie algebra $\mathfrak{g}$, I am trying to obtain a concrete realization of its simply connected Lie group $G$, with $\mathrm{Lie}(G) \cong \mathfrak{g}$. Let us ...
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1answer
25 views

Path Connectedness argument for $SO(n, \mathbb{R})$

I am trying to prove path connectedness of $SO(n, \mathbb{R})$. I have seen several different proofs for the same. But I had a thought and wanted to know whether it would help in any way. I took two ...
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1answer
45 views

Cover a sphere by two closed subsets not containing a closed self-antipodal connected subset?

Question (Fulton's Algebraic Topology, A First Course, Problem 4.40) Suppose the sphere $S^2=A\cup B$ where $A,B\subseteq S^2$ are two closed subsets of $S^2$. Is it true that either $A$ or $B$ must ...
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2answers
41 views

Connected Sets in Topology

Theorem: Let $(X,\mathscr{T})$ be a topological space. If $E$ is connected and $K$ is such that $E\subseteq K\subseteq\mathrm{cl}(E)$, then $K$ is connected. (Cl(E) is closure of E) Question: ...
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1answer
70 views

Is the sphere $S^n$ always arcwise connected?

I have a small question about the connectedness of the sphere; Is the sphere $S^n$ always arcwise connected ? Thank you.
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1answer
35 views

Simple question on connectedness in a subspace [duplicate]

For some reason I am having some trouble on this basic point set topology question: Suppose $X$ is connected, and $A$ is a connected subset of $X$, and that $B$ is a clopen set in $X-A$ (not in $X$, ...
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1answer
596 views

Proof: X is connected

Just came from an exam and I am wondering how to prove the following: A topological space $X$ is connected if for each continuous function $f:X\rightarrow X$ there is a $x \in X$ such that ...
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1answer
53 views

Connectedness and some properties [closed]

Let $S \subset \mathbb R^2 $ be defined by $$ S = \left\{ \left(m + \frac{1}{2^{|p|}}, n + \frac{1}{2^{|q|}}\right) : m,n,p,q \in \mathbb Z \right\}.$$ Is $S$ discrete? Is $\mathbb R^2\backslash S$ ...
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How to derive Jordan curve theorem for three arcs from the two arc version?

The Jordan curve theorem can be stated as follows: Let $p,q\in\mathbb R^2$, $p\ne q$ and $a,b$ be arcs between $p$ and $q$ intersecting only in the endpoints. Then $a\cup b$ divides $\mathbb R^2$ ...
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34 views

Is a plane minus a line a region?

According to ODE by Tenenbaum and Pollard, a region is defined as follows: Each point of the set is the center of a circle whose entire interior consists of points of the set. Every two points of ...
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27 views

Totally disconnected measurable set with positive measure

Could we find a totally disconnected set of the real numbers which is Lebesgue measurable and has positive measure?
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64 views

Prove that intersection of connected spaces is connceted.

Let A and B be connected subspaces of a topological space (X,$\tau$). If A,B are not disjoint, prove that the subspace A $\cap$ B is connected. Using the definition of connected space is that the ...
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1answer
51 views

Step Connected if and only if Connected

A space $X$ is step connected if given any open covering $\mathcal{U}$ of $X$ and any pair of points $p,q\in X$ there is a finite sequence $U_1,\ldots,U_n$ of sets belonging to $\mathcal{U}$ so that ...
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1answer
33 views

Connected components of the complement of a closed geodesic on a hyperbolic surface.

Let $M$ be homeomorphic to a 2-sphere with a finite number $\geq 3$ of points removed. This implies that $M$ can be equipped with a complete, finite area hyperbolic metric. I imagine $M$ as an ideal ...
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1answer
36 views

Number of connected components of the complement of a closed curve.

Let $\gamma:[0,1] \rightarrow \mathbb{R}^2$ be a continuous, closed curve (i.e. $\gamma(0) = \gamma(1)$). My question is about the number of connected components of the complement $\mathbb{R}^2 ...
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1answer
57 views

A maximal subset of $S^2$ with respect to a connectedness property

Let the set $A$ be a circle with a chord on the sphere $S^2$. Obviously $A$ has the following property: P: $\quad$ Any two points $a$ and $b$ of $A$ can be connected by a path that ...
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33 views

Disconnecting using totally disconnected sets [duplicate]

Let $X$ be $[0,1]^2$ and $S\subset X$ a totally disconnected subset. Is it true that $S^c$ is always connected? If it is false, what can we say when $X=[0,1]^n$?
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1answer
58 views

A question of topology.

If S is a subset of $\hspace{0.1cm}$$[0,1]\times[0,1]$$\hspace{0.1cm}$ such taht one point of the ordered pair is rational and the other is irrational or both are irrationals,then which of the ...
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1answer
43 views

Is for open connected $U$ the set $U_\varepsilon$ for small $\varepsilon$ connected?

Let for an open connected subset $U\subset \mathbb R^n$ and a number $\varepsilon >0$: $$ U_\varepsilon=\{x\in U: dist (x, \partial U)> \varepsilon \}. $$ Then $U_\varepsilon$ is open but in ...
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1answer
24 views

Need help understanding proof about connected sets.

I am struggling with the very last line of this proof: I mean, where does he get the existence of $z_1$ from?, and all the info about it?
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53 views

Homotopy of Involutory Matrices?

I want to construct a homotopy from the matrix $$ \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 1\\ \end{bmatrix} $$ ...
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Closed and Connected subgroups of $\mathbb{R}^n$

Question is : What are closed connected subgroups of $\mathbb{R}$ and from that deduce what are closed connected subgroups of $\mathbb{R}^n$ What i have done so far is : Only connected subsets of ...
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1answer
37 views

Product of Connected Spaces (2)

If $Y$ and $Z$ are connected, is $Y \times Z$ path connected? I cannot find a counter example. Some help please.
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1answer
61 views

Connected bipartite graph

Let $G$ be a bipartite graph with $n$ vertices. Prove that if every vertex has degree at least $\frac{n}{4} + 1$, then $G$ is connected. I'm assuming that number of vertices in this bipartite graph ...
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58 views

When does connectedness imply path-connectedness

In a locally path-connected space connectedness and path connectedness are equivalent. What is the minimal condition we would impose on a topological space to get the same result?