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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5 views

Prove that a edge-modified graph G' is connected if G contains a circuit

The following question was on my Discrete Math practice final: Let $G$ be a simple connected graph and $C$ a circuit. Let edge $e$ be in $C$. Prove that $H$ = $G-e$ = $(V, E-{e})$ by deleting $e$ ...
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0answers
17 views

Prove that the union of two given subsets of $\Bbb{C}^n$ is path-connected

Consider a subset $A$ of $Z=(\Bbb{C}^n$, Zariski topology) and regard it as a subspace of ($\Bbb{C}^n$, Metric topology). Sine $\Bbb{C}^n$ is homeomorphic to $\Bbb{R}^2n$, we can decide if A is ...
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2answers
34 views

Closed Euclidean ball in 1D and the Brouwer fixed point theorem

I'm slightly unsure about how the theorem is presented to me in lecture... The Brouwer fixed point theorem for $1$ dimension. Every continuous map $[0,1] \to [0,1]$ has at least one fixed point. ...
2
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1answer
31 views

Why is $(-\infty, \sqrt{2}) \cap \mathbb{Q}$ open in $\mathbb{Q}$?

I am looking at an example of disconnected spaces. Let $U=(-\infty, \sqrt{2})\cap \mathbb{Q}$ and $V=(\sqrt{2},\infty) \cap \mathbb{Q}$. Then $U,V$ are open in $\mathbb{Q}$ by definition of the ...
2
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3answers
38 views

What is the logic underlying this proof?

Proposition: A metric space $X$ is connected if, and only if, every continuous function $f:X\to (\{0,1\},d_D)$ is a constant function, where $d_D$ is the discrete metric on the set $\{0,1\}$. ...
0
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1answer
21 views

In Graph to tree: name of operation where edges removed and vertex/edge additions?

The graph has tree paths IN-1-OUT, IN-2-OUT and IN-3&4-OUT between IN and OUT in the left. I want to make each path to a branch like the right. What is the name of this operation or the name ...
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2answers
137 views

When can we recover a topology from its connected sets?

Definition. Let $X$ denote a set. Whenever $\tau$ is a topology on $X$, write $\tilde{\tau}$ for the collection of subsets of $X$ that are connected from the viewpoint of the space $(X,\tau)$. In ...
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1answer
19 views

Prove that in a 2-connected graph like G which has the vertex $v$, $v$ has a neighbor $u$ such that $G-v-u$ is connected

A graph $G$ is said to be $k$-connected (or $k$-vertex connected, or $k$-point connected) if there does not exist a set of $k-1$ vertices whose removal disconnects the graph. Let $v$ be a vertex of ...
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4answers
32 views

Problems understanding “connectedness”

I'm starting a foray into complex analysis, and I've come across the term "connected." I've vaguely heard the term before, but the book ("Complex Variables and Applications, 9th edition" by Brown and ...
2
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2answers
26 views

Covering map between two path connected sets

First off, I see a lot of variations of this problem cropping up on practice qualifiers, and I'm trying to regain my knowledge of topology. Let $p: X \to Y$ be a covering map where $X$ and $Y$ are ...
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1answer
63 views

How many connected components are left after removing a line from the plane?

Let $A \subseteq \mathbb{R}^2$ be a subset of the plane which is homeomorphic to $\mathbb{R}$. How many connected components does $\mathbb{R}^2 \setminus A$ have? My conjecture is that only one or ...
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1answer
46 views

A question about closed (but not necessarily compact) connected subsets of Euclidean spaces.

Is the following statement true?...... If $C$ is a non-degenerate closed and connected subset of the Euclidean plane $\mathbb R ^2$ and $p$ is any point of $C$, then there exists a connected ...
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1answer
37 views

Topology, Sorgenfrey [closed]

What are the connected subsets of R under the Sorgenfrey topology ? I actually cant find anything.. I tried several ways, but I got nothing.
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1answer
18 views

Connected, disconnencted, and homeomorphism

Say X = [0,1] T be the subspace topology induced on X, coming from the Standard Topology on R $$ Y = S^{1} $$ and T' be the subspace topology on Y, coming from the Standard Topology on $$ R^{2} $$ I ...
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2answers
66 views

Proof of $[0,1]~\text{disconnected}\implies(0,1)~\text{disconnected}$ [duplicate]

I want to prove the following implication $$[0,1]~\text{disconnected}\implies(0,1)~\text{disconnected}.$$ My try: Suppose $[0,1]=U\cup V$ with $U,V$ open, disjoint and nonempty. Using the subspace ...
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0answers
14 views

Union of the unit circle $S^1$ and the curve is connected but not path-connected

Prove that the union of the unit circle $S^1$ and the curve $W=\{(x, y) \in \mathbb{R^2} | x=(1-e^{-t})cost, y=(1-e^{-t})sint, t \geq 0\}$ is connected but not path-connected A connected space is ...
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1answer
40 views

If $\Omega$ is an open set of $\mathbb{C}$, $f$ constant on each connected component then $f$ is continuous

Let $\Omega$ be an open set of $\mathbb{C}$ and $f$ a constant function on each connected component of $\Omega$. I need to proof that $f$ is continuous. I've tried using that connected components ...
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0answers
49 views

Is it possible to prove this using set theory only and no more?

Got to prove: If $E$ and $F$ are connected sets, and $A$,$B$ are subsets of $E$ and $F$ respectively (but neither $A$ or $B$ are empty or fill $E$ and $F$entirely). Then $(A\times B)^c$ is connected ...
0
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1answer
26 views

An equivalent definition of connectedness in General Toplogy

I'm studying General Topology by the book of James Munkres and there he defined connectedness this way: "Let X be a topological space. A separation of X is a pair U, V of disjoint nonempty open ...
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1answer
26 views

Prove that In a locally connected space X every component U has the property that {U,X - U} forms a separation of X. [closed]

P rove that In a locally connected space X every component U has the property that {U,X-U} forms a separation of X.
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1answer
32 views

Prove that $W \cup S^1$ is connected in the subspace topology of $\mathbb{R^2}$

I want to solve the following question: Prove that the union of $W$ and the unit circle $S^1$ is connected in the subspace topology of $\mathbb{R^2}$ where $W=\{(x, y) \in \mathbb{R^2} | ...
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2answers
31 views

$\{(x, y) \in \mathbb{R}^2 \mid y=0\}\cup\{(x, y) \in \mathbb{R}^2 \mid x>0, y=1/x\}$ is not connected

This is an example from Munkres' book: $X=\{(x, y) \in \mathbb{R}^2 \mid y=0\}\cup\{(x, y) \in \mathbb{R}^2 \mid x>0, \ y=\frac 1 x\}$. Why is $X$ not connected? The author explained that neither ...
5
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1answer
30 views

Is $\mathbb{Q}^2 \cup \mathbb{I}^2 $ disconnected?

My intuition says that it is but I'm not entirely sure. I thought about using the projection map since it is continuous and surjective and also because I know that the rationals and irrationals are ...
0
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1answer
19 views

Prove topological space is connected

Prove : A topological space $X$ is connected if and only if the only continuous functions form $X$ into the discrete space $Y={0,1}$ are the constant functions, $f(x)=0$ or $f(x)=1$.
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1answer
23 views

Euler's formula about graphs embedded in $\mathbb{R^2}$

State and prove Euler's formula about graphs embedded into $\mathbb{R^2}$ I know that if we suppose $ G $ is a finite connected graph drawn on the surface of a sphere $ S^2 $. Then the ...
1
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1answer
36 views

Is the Cartesian product of irrationals disconnected in $\mathbb{R}^2$ disconnected?

If we consider $\mathbb{I}^2$ in $\mathbb{R}^2$then is this set disconnected? My intuitive guess is that it would be since an uncountable number of elements is being removed. Due to the density of the ...
0
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1answer
58 views

Subspace union of connected components equivalent to continuous function to $\{0,1\}$

Let $A$ be a locally connected topological space and $B$ a subspace. I want to prove the equivalence of the following two statements: (i) $B$ is a union of connected components of $A$. (ii) ...
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2answers
36 views

Prove every continuous function f: M -> R is a constant function

Assume M has only a countable or finite number of points and M is connected. Prove that every continuous function f:M->R is a constant function on all of M. Here is what I have so far: If f: M->R ...
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1answer
50 views

Constructing an analytic function for non simple connected domains

This is strongly related with this question, asked by me here a couple of days ago. Thanks to the Riemann Mapping thm proof (for example L.V Ahlfors' proof), we know that if $G \neq \mathbb{C}$ is a ...
5
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2answers
233 views

Space which is connected but not path-connected

Consider the following two definitions: Connected : A topologiocal space X is connected if it is not the disjoint union of two open subsets, i.e. if X is a disjoint union of two open sets A and B, ...
3
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1answer
62 views

On the Riemann mapping theorem

Let's take the family of analytic one to one functions, $f:G\to \mathbb{C}$ (with $G\neq \mathbb{C}$ a region and $z_0\in G$ a fixed point) such that $|f|<1$, $f(z_0)=0$ and $f'(z_0)$ is a real ...
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0answers
9 views

connect and arcwise-connect in locally convex space

Let X be a locally convex vector space and let G be an open connected subnet of X. How to show that G is arcwise-connected? I only can show that G is path-connected but do not know why G is ...
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1answer
18 views

Connectedness of factors of connected product space [closed]

If $A\times B$ is connected, why does it immediately follow that $A$ and $B$ are connected as well? I know that all functions $A\times B\rightarrow \{0,1\}$ are constant, but how would I prove that ...
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1answer
42 views

Connected sets and separations

Respect to this Prove that a connected space cannot have more than one dispersion points. , I couldn't proof the first item: Suppose that $p$ is a dispersion point of $X$. $X∖\{p\}$ is totally ...
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1answer
157 views

Does every connected metric space , with more than one point , contains a path connected subset with more than one point ?

Does every connected metric space , with more than one point , contains a path connected subset with more than one point ? Is there any additional condition imposing which on the mother space will ...
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1answer
25 views

Help with a proof involving connected spaces and homeomorphism

I wasn't sure the best way to word this in the title description so sorry if it's a bit vague. I'm struggling with the following homework problem. Let $X$ be connected, and assume a homeomorphism ...
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1answer
24 views

A convex subset of normed vector space is path-connected

Let $(N, \|\;\|)$ be a normed vector space and $(X,\tau)$ a convex subset of $(N,\|\;\|)$ with its induced topology. Show that $(X,\tau)$ is path-connected, and hence also connected. What I have done ...
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0answers
24 views

Using continuous functions to $\{0,1\}$ to prove that $A\times B$ connected iff $A,B$ are connected [duplicate]

Let $A,B$ be topological spaces. I want to prove $A\times B$ connected iff $A,B$ are connected. I have seen different ways to prove this, but I must use the hint: Use continuous functions to ...
1
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1answer
37 views

Topological space $\{a,b\}$ with topology $T=\mathcal{P}\backslash\{b\}$ path connected

I have the topological space $A=\{a,b\}$ of two elements with the topology $T=\{\emptyset,\{a\},\{a,b\}\}$. How do I prove that this is path connected? I know that path connectedness means that ...
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0answers
30 views

Is the complementary of a 2 codimensional set in a complex space simply connected?

It is well-known that if X is a complex manifold and $A \subset X$ an analytic set of codimension at least $2$ then $\pi_1(X)=\pi_1(X\setminus A)$. Can we generalize this equality when $X$ is a ...
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0answers
14 views

How to prove following properties about connectedness?

Prove that interval $(a,b)$ is connected in $\Bbb R$ Prove that $\Bbb R$ itself is connected Prove that any $n$-dimensional closed hyper-rectangle in $\Bbb R^n$ is connected. These problems are in ...
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1answer
13 views

Union path connected disjoint subspaces, $A, B$ and $C$, $X=A\cup B\cup C$ is path connected

Problem: Let X be a topological space and let $A, B$ and $ C$ be path-connected subspaces of $X$. Show that if $A \cap B \neq \emptyset$ and $C \cap B \neq \emptyset$ then the union $A \cup B \cup C$ ...
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0answers
19 views

Shrinking some polygons to make the containing polygon connected

Inside a public area $C$ (a polygon), there are several private land-plots $C_1,\dots,C_n$ (pairwise-disjoint simple polygons): Currentlly, the public area that is outside the private land-plots ...
0
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1answer
25 views

How to prove $\{(x,y) \in \mathbb{R}^2 \mid x^2+y^2<1\} \cup \{(x,y) \mid y=1\} \subset \mathbb{R}^2$ is path connected?

How to prove $\{(x,y) \in \mathbb{R}^2 \mid x^2+y^2<1\} \cup \{(x,y) \mid y=1\} \subset \mathbb{R}^2$ is path connected ? I could show above space is connected, but how to show path connectedness ...
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1answer
32 views

Deleted comb space proof

I'm trying to prove that the deleted comb space, $X$, is not path connected. We've defined $A = [0, 1] \times \{0\} \cup \{(\frac{1}{n}, t) : n \geq 1, t \in [0, 1]\}$, $X = A \cup \{(0, 1)\}$. From ...
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1answer
68 views

Using connectedness to show that $\mathbb{R}^n$ is not homeomorphic to $\mathbb{R}$

I want to show that $\mathbb{R}^n$ is not homeomorphic to $\mathbb{R}$, for all $n \geq 2$ using the following hint: Consider connected sets in $\mathbb{R}^n$ to $\mathbb{R}$, but things left in ...
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0answers
23 views

Quotients by simply connected closed subgroups

I have come across an exercise asking for a proof of something that is definitely false: If $G$ is a Lie group, $H$ a connected closed subgroup and $G/H$ simply connected, then $G$ is itself simply ...
5
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1answer
51 views

Every finite connected space is also path-connected?

Let $X$ be a connected space, if $X$ is finite, then $X$ is a path-connected space? If so, how to prove it? If not, how to give a counterexample? Thanks in advanced.
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0answers
13 views

Are only simple paths homotopic to a point? i.e. with winding number $1$

Title asks the question quite well. Is it only that curves with winding number $1$ which are homotopic to a point? Or is it really dependent on whether your space is simply connected? Or is this ...
3
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3answers
97 views

Is the set $\{\big(x,\sin(1/x)\big):x\ne 0 \}$ connected in usual metric of $\mathbb R^2$? [duplicate]

Is the set $\{\big(x,\sin(1/x)\big):x\ne 0 \}$ connected in usual metric of $\mathbb R^2$ ? I tried writing it as a union of two connected sets , or otherwise as a union of two disjoint non-empty ...