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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12 views

A connected locally compact metric space is sigma-compact without AC

Is it possible to prove that a connected locally compact metric space is sigma-compact without using the Axiome of Choice? Thank you.
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3answers
30 views

On the existence of connected metric spaces with open balls not connected

Does there exist a connected metric space with more than one point such that it has an open ball which is not connected ? Moreover does there exist a connected metric space with more than one point ...
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1answer
46 views

Does every compact simply-connected subset of $\mathbb{R}^n$ have an efficient $r$-covering path for all $r>0$?

Let $A$ denote a subset of $\mathbb{R}^n$. Definition 0. Given a positive real number $r$, an $r$-covering path of $A$ is a non-negative real number $T$ together with a differentiable function ...
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0answers
28 views

Connectedness of automorphism group of a variety

Let $Y$ be a proper, smooth, integral variety over an algebraically closed field $k$ of characteristic zero. Consider the automorphism group $Aut_k(Y)$ (a group scheme). Are there any natural ...
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66 views

How to find a path in $SL(2)$?

$SL(2,\mathbb{R})$ is path-connected. Therefore, for all $A,B \in SL(2,\mathbb{R})$ there is a path $\varphi:[0,1]\rightarrow SL(2,\mathbb{R})$, connecting both matrices. I would like to know, how I ...
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3answers
48 views

Is $E$ path connected $\implies \overline{E}$ connected?

Let $E\subset \mathbb R^n$ a path connectedness open set. Is $\overline{E}$ connected ? (where $\overline{E}$ is the closure of $E$). I tried to prove that it's true, but I don't get anything, may be ...
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0answers
38 views

Is $\mathbb{G}_{m,k}$ (the multiplicative group) simply connected?

I have a field $k$ (which I can take to be algebraically closed if it makes the answer simpler) with the char $k = 0$. The multiplicative group $\mathbb{G}_{m,k}$ is $spec (k [x, x^{-1}])$. ...
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2answers
35 views

Characterizing uncountable connected topological spaces

We know that if $X$ is a connected metric space with more than one point , then $X$ is uncountable ; can we characterize those connected topological spaces for which more than one point implies ...
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1answer
22 views

Characterizing connected sets of $\mathbb R^n$ is terms of differentiable maps for which zero derivative everywhere implies constant

Let $U $ be an open subset of $ \mathbb R^n$ ; then how to prove that $U$ is connected iff for every differentiable function $f:U \to \mathbb R$ , $\nabla f(x)=0 \implies f $ is constant on $U$ ?
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2answers
28 views

continuous map of connected set is connected, example: Proving the connectedness of this set.

I thought I would try to use this to prove connectedness in this set if possible: $$\{(x,y)\mid 1<x^2+y^2<4\}$$ $f(x,y)=x^2+y^2$ So since $(1,4)$ is connected in $\mathbb R$ so it this set, as ...
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61 views

Prove $\{(x,y): x>0\}$ is connected

As an introduction to multivariable calculus, I'm given a small introduction to some topological terminology and definitions. As the title says, I have to prove that $\{(x,y): x>0\}$ is connected. ...
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1answer
76 views

Show that $S$ is connected

Let $S=\{x\in\mathbb R^n:||x||=1\}$ with $ n>1$. Show $S$ is connected without using arcwise connectedness. I would be done if I can show this: Let $X$ be a connected space and $A\subset X$ be ...
4
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1answer
59 views

Is $f$ constant when each point is local minimum or maximum?

Let $X$ be a connected topological space and $f:X\to \mathbb R$ be continuous. Further, we know that all $x\in X$ are local extrema. Does that imply that $f$ is constant? I think in case $X$ is ...
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1answer
21 views

$X$ be a complete metric space and $f:X \to X$ be a bijective and connected preserving map ; then is $f^{-1}$ also connected preserving?

Let $X$ be a complete metric space and $f:X \to X$ be bijective and a connected preserving map i.e. $f$ carries every connected set of $X$ to a connected set of $X$ ; then is it necessarily true that ...
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1answer
52 views

Rudin 4.22 Let $f$ be a continuous function from $X$ to $Y$ (metric spaces). If $E$ is connected, then $f(E)$ is connected

Rudin 4.22. Let $f$ be a continuous function from $X$ to $Y$ (metric spaces). If $E$ is connected, then $f(E)$ is connected. Could someone check this proof: Proof: I will show the ...
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1answer
38 views

Trouble understanding path-connected components of the topological space C(X,Y)

$C(X,Y)$ is the set of all continuous functions from topological space $X$ to topological space $Y$. I'm trying to prove this theorem: If $X$ is locally compact, two functions $f,g \in C(X,Y)$ are in ...
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1answer
27 views

Open connected subsets of path-connected spaces

Let $X$ be a path-connected topological space and $Y$ an open connected subset. Is $Y$ path-connected?
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37 views

$1 \leq a^2x^2 + b^2y^2 - abxy \leq 9 , x\geq 1$- question compactness and connectedness..

I was told that this object was a cone, I cannot see that, can anyone tell me how to identify which object this is, so as to continue assesing and answering questions of compactness and ...
2
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2answers
57 views

Whether the set $A$ , $ A=\{y\in R : y= \lim_{n\rightarrow \infty} f(x_{n}) \text{ where } x_n \text{ diverges to } +\infty \}$ is connected

Let $f: \mathbb R \rightarrow \mathbb R$ be continuous function and $A\subset \mathbb R$ be defined by $$ A=\{y\in R : y= \lim_{n\rightarrow \infty} f(x_{n}) \text{ where } x_n \text{ diverges ...
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49 views

General topology exercise (equivalent condition for simple connection)

Let $X$ be a pathwise-connected topological space. Prove that $X$ is simply connected iff every continuous $f:S^1\to X$ can be extended to a continuous function $g:D^2\to X$. How can I use the fact ...
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53 views

$G =(V,E)$ is $k$-connected ($k \geq 2$), prove that for every subset $S \subseteq V $, |S|=k there exists a cycle in $G$ that goes through all of $S$ [duplicate]

I thought of starting from the Menger theorem which says that between every two vertices $u$ and $v$ there are $k$-edge disjoint graphs. So I think if I look at $G$ without the subset $S$ then I have ...
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87 views

Is the inverse of a bijective connectedness preserving map , on a complete real inner product space , also connectedness preserving?

Let $X$ be a complete real inner-product space and $f:X \to X$ be a bijection which maps connected sets to connected sets ; then is it necessarily true that $f^{-1}$ also maps connected sets to ...
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1answer
30 views

If $A\subseteq\Bbb R$ is nonempty with $|A|\ge 2$, then $A$ totally disconnected $\iff A^\circ=\emptyset$.

In the course of working on an exercise, I came up with the claim given in the title. Just looking for verification. $\underline{\text{Claim: } A\text{ is totally disconnected}\iff ...
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1answer
25 views

Connected components and showing subsets are equal

If $Z_1, Z_2 \subset X $ are connected components, show that $Z_1 = Z_2$ or $Z_1 \cap Z_2 = \emptyset$ Note: we defined connectedness as a splitting of two open sets $U_1$, $U_2$ such that$U_1 = X$ ...
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0answers
16 views

Find Connected Components

Let $X = \mathbb{R^2}$ and let $F_k$ be the closed line segment joining $(-1,2^{-k})$ and $(1,2^{-k})$ Additionally let $a = (-1,0) \ ,\ b=(1,0)$ Consider $A = \{a,b\} \cup \ ...
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1answer
22 views

Prove this is a Connected Component

Let $Y$ = $ \{(x,y) \in \mathbb{R^2}\ |\ xy = 0\ \ \cap\ (x,y) \neq (0,0)\ \}$ Consider $U_+ = \{\ (x,0) \in \mathbb{R^2}\ |\ x>0\ \}$ I'm trying to show that $U_+$ is a Component of $Y$ My ...
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1answer
49 views

Prove that $\{(x,y)\mid x\in\mathbb Q\}\cup\{(x,y)\mid y\in\mathbb Q\}$ is a connected subset of $\mathbb R ^2.$ [duplicate]

In my notebook, something is very briefly, not in detail whatsoever, path connectedness mentioned, and two assumptions are made about $x_1, x_2 \in \mathbb Q.$ If anyone can prove this I would greatly ...
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4answers
78 views

Proving that a continuous $f:X \to Y ; \ X,Y- \text{topological spaces}$ and $A \subseteq X ; A \text{ connected} \implies f(A) \text{ connected}$

Proving that $f:X \to Y ; \ X,Y- \text{topological spaces}$ and $A \subseteq X ; A \text{ connected} \implies f(A)-\text{connected}$ The answer is given like this just one step I do not understand ...
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2answers
87 views

A ring is a connected set

I not know how to prove this: For example $$A=\{(x,y,z)\in \mathbb{R^3}\mid 1 < x^2 + y^2 + z^2<2 \}$$ I know that $$\partial A=S(0,1)\cup S(0, \sqrt{2})$$ can that help me at all? I was also ...
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1answer
72 views

Is simply connectedness is a topological property?

A topological space $X$ is called simply-connected if it is path-connected and any continuous map $f:S^{1} \to X$ (where $S^1$ denotes the unit circle in Euclidean 2-space) can be contracted to a ...
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1answer
65 views

Locally connected Locally compact separable metric space

Let $X$ be a locally connected locally compact separable metric space. Is it possible to find a countable collection $\mathcal{B}$ such that every member of $\mathcal{B}$ is a nonempty peano subspace ...
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1answer
48 views

With what additional assumption, would a connected space be path connected?

Let $X$ be a connected space. What additional condition on $X$ would imply that $X$ is path-connected? The only one I know is by assuming $X$ is an open subspace of a normed space $V$. What else ...
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1answer
65 views

$X$ is locally connected and $f: X \to Y$ is onto where $Y$ has the quotient topology. Prove that $Y$ is locally connected

Those questions about connectivity drive me crazy - I'm having so much difficulty proving them. Say $X$ is a locally connected space, and $f: X \to Y$ is onto where $Y$ has the quotient topology. ...
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2answers
70 views

Connected Lie group is second countable?

I know this is true from various sources, unfortunately none of them give the full proof. I already have a start: Let $G$ be connected Lie Group. Choose $K$ to be any compact neighbourhood of the ...
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1answer
96 views

$X$ is connected, $A \subset X$ connected, and $C$ a component of $X\backslash A$. Is $\overline A \cap \overline C \ne \emptyset$?

I'm trying to prove or disprove the following statement: If $X$ is connected, $A \subset X$ is connected, and $C$ a component of $X\backslash A$ then $\overline A \cap \overline C \ne \emptyset$. I ...
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1answer
44 views

Showing pre-image under entire function is simply connected.

I am currently working on the following problem and have run into a bit of trouble: Consider an entire function $f$ s.t. $\overline{B_1(0)}\subset f(\mathbb{C}).$ Show that V, a component of ...
3
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1answer
48 views

Prove that if the closure of each open ball in compact metric space is the closed ball with the same radius, then any ball in this space is connected

I'm having some difficulty with the following problem in general topology: Prove that if the closure of each open ball in compact metric space is the closed ball with the same center and radius, then ...
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34 views

If base+fibres are connected, is the total space connected?

I have a morphism $f: X \to Y$ of smooth varieties over an alg. closed perfect field $k$. If $Y$ is connected and all the fibres of $f$ are connected, does it follow that $X$ is also connected? ...
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90 views

An open and connected subset $U\subseteq \mathbb C$ is still connected if you remove a curve that lies entirely in $U$

Let $U\subseteq \mathbb C$ be open and connected. If $f:[0,1]\rightarrow U$ is continuous with $f(0)\neq f(1)$ and $f(s)\neq f(t)$ for $s\neq t$, then $U\setminus f([0,1])$ is connected. This seems ...
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1answer
36 views

Unit quaternion ball is compact and connected?

Let$$\mathbb{U} := \{x \in \mathbb{H} : |x| = 1\}.$$This is a group under multiplication. What is the easiest way to see that $\mathbb{U}$ is a compact and connected subset of $\mathbb{H}\cong ...
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1answer
32 views

Hausdorff space definition in terms of disconnected subsets

I've read the definition of a topological Hausdorff space (two distinct points have disjoints neighbourhoods) and of a disconnected space (it is the union of two disjoint nonempty open sets) Now I ...
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66 views

$R^2\setminus K, K$ compact, is not simply connected

So, I think I'm aware of the general idea of how to do this. $K$ is compact, hence closed and bounded, so there is some circle of finite radius, say $r$, that wraps around $K$. This loop isn't null ...
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0answers
35 views

When does “every closed path is homotopic to a point” imply the space is path connected?

In the middle of talking about primitives and the Cauchy integral theorem, my Complex Analysis teacher came up with this sentence: This reasoning can be done in any simply connected set, because ...
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1answer
23 views

Counting weakly connected graphs with outdegree of exactly one.

If we count all graphs of $N$ labelled vertices, where each vertex has an outdegree of exactly $1$ with no self-loops allowed, we'll find that there are exactly $(N-1)^N$ of them (for every of $N$ ...
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1answer
19 views

For every $n \ge 2 $ , existence of uncountably many mutually disjoint closed balls , complement of whose union is path connected

For every $n \ge 2 $ , does there exist an uncountable family $\{D_\lambda: \lambda \in \Gamma\}$ of mutually disjoint closed balls in $\mathbb R^n$ , such that $\mathbb R^n \setminus \cup_{\lambda ...
2
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1answer
41 views

Is this argument invalid?

So, what I was trying to prove is: Let $\pi : X \to Y$ the quotient map such that $\pi^{-1}(\{y\})$ connected for all $y \in Y$. Then $X$ is connected. Suppose yet that $X$ is locally connected. ...
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157 views

Show the following set is connected

For any $x \in \Bbb R^n$ how do I show that the set $B_x := \{{kx\mid k \in \Bbb R}$} is connected. It should also be concluded that $\Bbb R^n$ is connected. I was thinking of starting by assuming ...
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1answer
37 views

removing rationals in topolosist's sine curve

Munkres pg. 160 says, where $S$ is the topologist's sine curve If one forms a space from $\bar S$ by deleting all points of $V$ having rational second coordinate, one obtains a space that has only ...
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0answers
34 views

Other counterexamples for this problem on connectedness

If $A$ is a connected subspace of $X$, does it follow that $\operatorname{Int}A$ and $\operatorname{Bd}A$ are connected? Does the converse hold? Justify your answers. For original problem, I can ...
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2answers
25 views

Is this subset of the plane path connected?

Inspired by this question I asked myself the question which I am going to describe: Let $\mathbb {I}$ be the set of all irrational numbers. Let $\mathbb {I}^2$ be the Cartesian product of the set ...