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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2
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1answer
22 views

If $A$ is a component in $(X,d_1) \iff$ $A$ is a component in $(X,d_2)$ then is $(X,d_1)$ homeomorphic to $(X,d_2)$?

Suppose that $A$ is a component in $(X,d_1) \iff$ $A$ is a component in $(X,d_2)$. Is it always the case that $(X,d_1)$ is homeomorphic to $(X,d_2)$? I have been trying to find a counter example, but ...
1
vote
2answers
51 views

Why is a circle not simply-connected?

To be simply-connected means to be path-connected and able to continuously shrink a closed curve while remaining in the domain. According to wikipedia, a circle is not simply connected, but a disk ...
2
votes
1answer
11 views

Find a connected graph that has exactly $2$ cutpoints of order $2$ and $3$ cutpoints of order $3$

Find a connected graph that has exactly $2$ cutpoints of order $2$ and $2$ cutpoints of order $3$ Definition: A cut point of order $k$ is a point $a \in X$ whose complement $X-\{a\}$ consists of ...
1
vote
2answers
39 views

Is this proof regarding product of connected spaces correct?

Let $X,Y$ be connected spaces, and consider their product $X\times Y$. I want to show that their product is connected. The posts I've read here regarding this question often include creating "slices" ...
1
vote
1answer
24 views

Why are the (connected) components of a topological space themselves connected?

I am trying to prove that (connected) components of a topological space are connected. I'll first define what I mean by a 'component of a topological space': For a topological space $X$, write ...
0
votes
1answer
28 views

Showing the Following Set is Path Connected

Would be grateful if someone could help me with the following question "For any normed space X which is NOT the set of Reals, the set X (excluding 0) is path connected In the solutions the usual ...
0
votes
0answers
21 views

Complex variable, multiplication of numbers

Question: Let a and b be complex numbers with $a \neq 0.$ Describe the set of points $az + b $ as $z$ varies over the first quadrant, $\{z = x+iy: x>0 \,and \,y>0\}$ Solution: Let $a = ...
1
vote
3answers
60 views

Does there exist a continuous function $g:S^1 \to S^1$ such that $(g(z))^2=z , \forall z \in S^1$?

Let $S^1:=\{z \in \mathbb C:|z|=1\}$ ; does there exist a continuous function $g:S^1 \to S^1$ such that $(g(z))^2=z , \forall z \in S^1$ ?
1
vote
2answers
27 views

Closed unit square is connected.

Question. Prove that the closed unit square $S=\{x$ in $\Bbb R^2 : 0 \le x_1 \le 1, 0 \le x_2 \le1 \}$ is connected. I understood how to prove unit interval is connected and I am trying to extend to ...
4
votes
1answer
54 views

Connected components functor for free coproduct cocompletions

Any extensive category admits a notion of connected object and hence a disconnected object. However, not all disconnected objects are presentable as disjoint unions of connected objects. Among ...
1
vote
1answer
33 views

For every pair $n,d$ such that $d \ge (n-1)/2$ prove that $G$ on $n$ vertices with minimum degree $d$ is edge d-connected.

For every pair $n,d$ such that $d \ge (n-1)/2$ prove that $G$ on $n$ vertices with minimum degree $d$ is edge d-connected. None of my observations I was able to obtain seem to be useful. I am just ...
1
vote
1answer
45 views

Vertex connectivity of $K_n$ upon removal of edges of subgraph $C_n$

Consider graph $G$ = $K_n$ - $E(C_n)$ ($G$ is complete graph on $n$ vertices upon removal of edges of subgraph $C_n$). For every $n \ge 3$ find maximum $k$ such that $G$ is vertex $k$-connected. I ...
1
vote
1answer
24 views

$X \subseteq M(n,\mathbb C) ; |X|>1 ; $ connected /path connected , what about $S:=\{x \in \mathbb C : x $ is an eigenvalue of some matrix in $X\}$?

Let $X \subseteq M(n,\mathbb C)$ be a set with more than one element ( I am also considering $M(n,\mathbb R) \subseteq M(n,\mathbb C)$ and $S:=\{x \in \mathbb C : x $ is an eigenvalue of some matrix ...
0
votes
1answer
41 views

“Continuous maps are those maps that do not tear space apart”

In a tutorial I wanted to give a quick explanation of the property of continuity. One of the common intuitions for continuity is that it preserves connection: Continuous maps do not map connected ...
2
votes
1answer
41 views

Connectedness of the sets $\{A \in M(n,\mathbb R) : \mathrm{rank}(A) =m\}$ and $\{A \in M(n,\mathbb R) : \mathrm{rank}(A)\ge r\}$

Let $r>0$ , I know that $\{A \in M(n,\mathbb R) : \mathrm{rank}(A) < r\}$ is path connected in $M(n,\mathbb R)$ . My question is ; for positive integer $m < n$ , is the set $\{A \in ...
1
vote
1answer
40 views

How does $\frac{1}{2}(n-s+1)(n-s)$ equal $\binom{n-s+1}{2}$?

Maybe a basic question, but I'm strolling through graph theory at the moment after a few years out of tertiary mathematics. There is a theorem that if a graph $G$ has $s$ connected components, then $$ ...
1
vote
0answers
50 views

On a function $f: \mathbb R^m \to \mathbb R^n$ , $n>1$ , mapping connected sets to connedted sets and discontinuous at a point

Let $f: \mathbb R^m \to \mathbb R^n$ be a function mapping connected sets to connected sets where $n>1$ ; let $a \in \mathbb R^m $ and $ \epsilon >0$ be such that $f(B_{\delta}(a)) \cap ...
2
votes
1answer
31 views

Locus of $xy=1$ is not path-connected

Artin, Algebra, Chapter 2, M6 It is intuitively easy but I am not sure if the writing my proof is rigorous enough. Let $a=(a_1,a_2,\cdots,a_k)$ and $b=(b_1,b_2,\cdots,b_k)$ be points in ...
8
votes
3answers
152 views

$f$ is an entire function such that $f(0)=0$

Let $f$ be a non-constant entire function satisfying the following conditions: $f(0)=0$ and for each $N \gt 0$ the set $\{z \mid \left| f(z)\right| < N\}$ is connected. Prove that $f(z)=cz^n$ for ...
1
vote
1answer
41 views

$A\times B$ connected component implies $A,B$ connect components

$X,Y$ topological spaces. I want to show that if $C\subseteq X\times Y$ is a connected component, then $C=A\times B$ where $A,B$ are connected components of $X,Y$. What I have so far, is that any ...
2
votes
1answer
23 views

Given that $H^1(X)=0$ on a connected space, show that all maps to $X\to S^1$ are null homotopic

Let $X$ be a path-connected, locally path-connected topological space, with $H^1(X)=0$. I would like to show that any map $f:X\to S^1$ is null homotopic, but I haven't really made any progress. ...
1
vote
1answer
26 views

Help with a proof; Path connected open subspaces imply $X$ to be path connected

This is a problem I cannot prove, Show that if $X$ is connected and is such that for every $x \in X$ there exists and open path-connected set $U$ with $x \in U$ then it is path-connected. Here's ...
1
vote
0answers
19 views

Is $\mathbb{R}^n$ connected under the density topology?

Let $\mu$ denote Lebesgue measure and define the function $ d : \mathbb{R}^n \times \text{PS}(\mathbb{R}^n) \to [0,1] $ by $$ d(x,S) = \lim_{r \to 0^+}{\frac{\mu(B(x,r)\cap S)}{\mu(B(x,r))}}. $$ We ...
5
votes
1answer
83 views

Property similar to connectedness

Recall that $X$ is connected if $X$ cannot be written as the union of nonempty open sets with empty intersection. Consider the following similar property: $X$ is good if $X$ cannot be written as ...
0
votes
0answers
7 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 $e$ ...
1
vote
2answers
35 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
votes
1answer
34 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
votes
3answers
40 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
votes
1answer
24 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 ...
13
votes
2answers
142 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 ...
0
votes
1answer
26 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 ...
0
votes
4answers
37 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
votes
2answers
28 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 ...
4
votes
1answer
65 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 ...
1
vote
1answer
49 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 ...
-3
votes
1answer
42 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.
0
votes
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 ...
0
votes
2answers
70 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 ...
0
votes
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 ...
0
votes
1answer
41 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 ...
1
vote
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
votes
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 ...
-1
votes
1answer
27 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.
6
votes
1answer
82 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} | ...
0
votes
2answers
32 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
votes
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
votes
1answer
20 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 $f(x)=1$.
0
votes
1answer
25 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
vote
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
votes
1answer
59 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) ...