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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2answers
23 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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1answer
60 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
48 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
25 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
30 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 ...
5
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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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0answers
32 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
54 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 ...
3
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1answer
40 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
22 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?
3
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3answers
51 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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0answers
70 views

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 ...
2
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1answer
35 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.
2
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1answer
48 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 ...
4
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0answers
54 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?
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1answer
26 views

Good Methods for Proving Connectedness

Is there a good standard way of proving connectedness? Disconnectedness seems a lot easier, as you just need to find the sets. For connectedness is there a similarly straightforward approach? I think ...
3
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1answer
47 views

$A\cup B$ is connected when $A$ is connected in $X$ and $B$ clopen in $X-A$

Let $A$ be connected subset of a connected space $X$, and $B\subset X-A$ be an open and closed set in the topology of the subspace $X-A$ of the space $X$. Prove that $A\cup B$ is connected. I think ...
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0answers
39 views

Prove: If H and G/H are totally disconnected then G is also totally disconnected

Let $G$ be a topological group and $H$ a subgroup of $G$. If $H$ and $G/H$ are totally disconnected, then $G$ is also totally disconnected. With 'totally disconnected' we mean the every connected ...
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1answer
24 views

Proving a region is pathwise connected

I am having problems trying to prove the following statement: Let $\Omega \subset \mathbb C$ be a region (i.e., an open, nonempty, connected subset of $\mathbb C$). Prove that for all $z_0,z_1 \in ...
1
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1answer
67 views

Countable subsets are disconnected

I am trying to show the following: every countable subset of $\Bbb R$ with at least two points is disconnected. My attempt: let $D$ be such subset. Then take $a \in D$ and define $A=\{ a\}$ and $B = ...
2
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1answer
66 views

is $(\mathbb{Q} \times (\mathbb{R}\setminus\mathbb{Q}))\cup((\mathbb{R}\setminus\mathbb{Q})\times\mathbb{Q})$ connected? path connected?

let $$X=(\mathbb{Q} \times (\mathbb{R}\setminus\mathbb{Q}) ) \cup ((\mathbb{R}\setminus\mathbb{Q})\times\mathbb{Q}) $$ and let $$\tau=\tau (\text{euclid})$$ what are the connected components of ...
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1answer
60 views

Proof: The quotient space is totally disconnected

I want to prove the following: Let $X$ be a topological space. Remark: $x \sim y \iff$ There is a connected component which contains $x$ and $y$. And now I want to show that the quotient space ...
3
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1answer
29 views

Show that $Y$ is not path-connected

Let $\mathbb{R}^2$ with the usual topology and let $$ Y = A_0 \cup (\bigcup_{n \in \mathbb{N}} A_n) \cup (\bigcup_{n \in \mathbb{N}}L_n)$$ where $$ A_0 = \{ 0 \} \times [0,1] \qquad A_n = \{ ...
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2answers
103 views

Arcwise connected but not connected?

In his book "Geometry, Topology and Physics", Nakahara makes the following claim with regard to topological spaces: With a few pathological exceptions, arcwise connectedness is practically ...
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1answer
101 views

Path-connected and locally connected space that is not locally path-connected

I'm trying to classify the various topological concepts about connectedness. According to 3 assertions ((Locally) path-connectedness implies (locally) connectedness. Connectedness together with ...
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1answer
18 views

Let $K_1(0) := \{x \in \mathbb{R}^2: \|x\|_2 < 1\}$ and $S := K_1(0) \setminus \mathbb{Q}^2$. Is M path connected?

The Assignment: Let $K_1(0) := \{x \in \mathbb{R}^2: \|x\|_2 < 1\}$ and $S := K_1(0) \setminus \mathbb{Q}^2$. Is S path connected? Explain your answer. I don't think S is path-connected since ...
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1answer
30 views

connectivity on graph

Given a graph, undirected or directed, what is the optimal or just good algorithm for finding the following? 1) Whether two vertices are connected. 2) The shortest path going from one to the other. ...
0
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1answer
124 views

Hyperspace and connectedness

I'm looking for any theorems and proofs for connectedness for hyperspaces exp(X). I would like to take a look for especially this theorem: $$ X \textit{ is connected } \leftrightarrow exp(X) ...
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1answer
65 views

Number of connected components of this complement

Let $X$ be a locally finite simplicial complex and let $K$ be a finite subcomplex of $X$. Why is the number of connected components of the complement $X-K$ finite?
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0answers
64 views

Connectedness of parts used in the Banach–Tarski paradox

A quote from the Wikipedia article "Axiom of choice": One example is the Banach–Tarski paradox which says that it is possible to decompose the 3-dimensional solid unit ball into finitely many ...
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0answers
49 views

Does $\pi_0$ commute with sequential colimits?

I'm looking for an example of a sequence of topological spaces $Y_1 \rightarrow Y_2 \rightarrow \cdots$ such that the induced map $$ \text{colim}\, \pi_0(Y_i) \rightarrow \pi_0 (\text{colim}\, Y_i) $$ ...
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1answer
46 views

Finitely many connected components, prove interiors are also connected

Show that in a space with finitely many connected components $C_i, i = 1, ..., n$ their interiors $Int(C_i)$ are also connected. Is it true in general that the interior of a connected component is ...
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3answers
72 views

If n > 1 and $B \subset \mathbb R^n$ countable. Then $\mathbb R^n - B$ is connected ( James Dugundji)

We can assume that $0 \in B$ , otherwise we move the origin.. We show that the origin and $ x \in \mathbb R^n - B$ are contained in a connected set lying in $\mathbb R^n - B$. Draw ...
3
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2answers
189 views

Countable basis but uncountably many connected components

Looking for some guidance on two topology questions: (a) Show that a locally connected space with a countable basis, has at most countably many connected components. (b) Give an example when X has ...
0
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1answer
26 views

Show that set is path connected?

How do I show that the set $A = \{(x,y) \in R^2: x \geq 0, y \geq 0\} \cup \{(x,y) \in R^2: x \leq 0, y \leq 0\}$ is path connected. I know that I need to construct a continuous function $f:[0,1] ...
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1answer
32 views

Finding the connected components of topological spaces

Find the connected components of the following sets: $(a) \; A=\{(x,y):y=\sin(1/x), x\in\mathbb{R}^+\},(b)\;A\cup\{(x,y):x=0,y\in[-1,1]\},(c)\;$The Cantor Set,$(d)\;\mathbb{N}$ with the cofinite ...
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1answer
33 views

How to show $A=\{(x,y)\in R^2:4x^2+9y^2=36\}$ is path connected and compact?

let $A=\{(x,y)\in R^2:4x^2+9y^2=36\}$ . Show that A is path connected and compact. my attempt: since $\frac {x^2}{9}+\frac{y^2}{4}=1$ is elips. A is bounded and closed. so is compact. (by heine ...
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1answer
42 views

How to show “ If A and B connected, is $A\cup B$ connected”?

If A and B connected, is $A\cup B$ connected? or give a contrary example. I'd say no because when we take A=[1,2], B=[3,4]. this closed intervals are connected. but when we take $U=]\frac 12,\frac ...
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1answer
67 views

Deleting $n$ points from a connected space

Let $X$ be a space such that for any subset $S \subset X$ with finite cardinality $n$, the subspace $X \setminus S$ has exactly $n+1$ connected components, each of which is homeomorphic to $X$. Is ...
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3answers
201 views

Möbius band with its middle part removed is still connected

Let $I\times I/(0,t){\sim}(1,1-t)$ be the Möbius band and let $S=\{(x,y): (x,y)\in M, 1/4<y<3/4\}$ be its middle part. How can I show that $M-S$ is connected? I tried to write a continuous ...
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0answers
14 views

complement of complex submanifold connectivity

Let V be a closed complex submanifold of $\Bbb C\Bbb P_n $ of complex dimension $n-1$. Show that the complement of V in $\Bbb C\Bbb P_n $ is connected but need not be simply connected.
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6 views

discrete connected gird

1 2ave found the following statement without a proof: Given a set $ \Omega \subset \mathbb{R}^2$. Now let $G_h=\{ (x,y) \mid x/h \in \mathbb{Z} \quad or\quad y/h \in \mathbb{Z}\}$ be the standard ...
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1answer
24 views

Show number of spanning trees on a biconnected graph

How to show that a biconnected graph on $n$ vertices has at least $n$ different spanning trees? I just have no idea, how to solve a problem like this. Help please, any hint is appreciated.
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1answer
25 views

Can a connected subpsace be in disjoint open sets?

Say $X$ is a non-connected topological space, i.e. $X= U\cup V$, and $U,V$ are disjoint (non-empty) open sets. Then suppose $C$ is a connected subspace of $X$, with the standard subspace topology. Can ...
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0answers
30 views

Constant Function over Connected, Compact Space

I am working on this problem and was wondering if I could get some feedback on my attempt at the proof. My gut tells me that I need a stronger argument as why my covering is actually a cover. I also ...
1
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1answer
136 views

Connectedness problem on the 2-sphere

Let $K \subset L \subset S^2$, where $S^2$ is the 2-sphere and $K$ and $L$ are compact subsets with empty interior and $L$ is connected (I don't think all of those are relevant hypotheses though, but ...
0
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1answer
30 views

Examples of continuous integer-valued functions on totally disconnected spaces

I wanted to see examples of continuous integer-valued functions $f:X\to \mathbb{R}$ on a totally disconnected space $X.$ I have only some abstract examples in mind.
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0answers
20 views

let $p: E\rightarrow B$ conitnuous and surjective. Show that if U is connected,then the partition of $p^{-1}(U)$ into slices is unique.

let $p: E\rightarrow B$ coninuous and surjective and suppose that $U$ is an open set of $B$ that is evenly covered by p and U. Show that if U is connected,then the partition of $p^{-1}(U)$ into slices ...
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1answer
34 views

Compact and connected

Let $\mathbb{J} :=\{1/n: 0< n\in \mathbb{Z}\}$ Let $T_{ir}$ be topology of $\mathbb{R}$ generated by $$\{(a,b)\subset \mathbb{R}:a<b\}\cup\{(a,b) \setminus \mathbb{J}\subset ...
1
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1answer
35 views

Dugundji problem about quasicomponents.

Problem 5 part d), of chapter V, section 3 (p. 118): d. In $E^2$ let $L_1$ be the line $x=1$ and $L_2$ the line $x=-1$. For each $n\in\Bbb Z^+$ let $R_n$ be the rectangle $\{(x,y):|x|\le ...