Questions tagged [connectedness]

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

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Let $U \subset S^n$ open subset of the sphere homeomorphic to $\mathbb{R}^n$ prove that $\mathbb{S}^n \setminus U$ is connected

Let $U\subset S^n$ open subset of the sphere homeomorphic to $\mathbb{R}^n$ prove that $\mathbb{S}^n \setminus U$ is connected Assume that $\mathbb{S}^n \setminus U$ isn´t connected then there ...
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Show that $\{(0,0)\}$ and $\{(0,1,)\}$ are both connected components of $A$ [duplicate]

For each positive integer $n$ define a subset $A_n$ of $R^2$ by $A_n=\{1/n\}$ x $[0,1]$, and let $A=\cup_n A_n \cup\{(0,0),(0,1)\}$. Show that $\{(0,0)\}$ and $\{(0,1,)\}$ are both connected ...
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About disconnected set [closed]

We know that a non singleton connected set $A\subset \Bbb R$ must be an interval. Also, we know if $A$ is not connected then we say $A$ is disconnected. But I do not know what does it mean that $A$...
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Difficulty with Definition of Disconnectedness w.r.t. closed sets

Here's what I know: When is a (metric) space disconnected? Consider a metric space $(M,d)$. $M$ is disconnected if there exist non-empty disjoint open subsets $A,B \subset M$ such that $A\cup B = M$. ...
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Is $A=\left\{x:x \text{ belongs to } (1,2) \cup [2,3)\right\}$ connected?

My question is whether the given set $$A=\left\{x:x \text{ belongs to } (1,2) \cup [2,3)\right\}$$is connected? If so then how? I know $x$ belongs to $(1,2)$ implies that $1<x<2$ and $x$ belongs ...
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The torus is connected [duplicate]

Let $T^2=S^1 \times S^1$ be a torus, where $S^1 \subset \mathbb{R}^2$ is the unit circle. Is $T^2$ connected? If so, how does one show this?
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Open in which topology? Seifert–Van Kampen theorem

Is a particular case of Seifert-Van Kampen theorem that if $U$ and $V$ are open sets simply connected and $U\cap V$ is path connected, then $X=U\cup V$ is simply connected. I have dificulties with the ...
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Is the smoothness required in the following question?

Is the smoothness required in the following question ? I think continuity is sufficient to prove the function being continuous. My attempt: As $g$ is a smooth function , $g$ will also be a ...
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Find if $W=\lbrace z\in \mathbb{C}\mid 2 \leq |z| \leq 3 \rbrace$ set is connected and compact

Let $W=\lbrace z\in \mathbb{C}\mid 2 \leq |z| \leq 3 \rbrace$. Prove that $W$ is connected and compact. Let $z=x+iy,\: x,y\in\mathbb{R}$. Notice that $$W=\lbrace z\in \mathbb{C} \mid 4 \leq x^2+y^2 \...
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Can we say that $f$ is smooth?

Let $U \subseteq \Bbb R^k$ be an open subset which is not connected. Then there exists a smooth locally constant surjective function $f : U \longrightarrow \{0,1\}.$ What I know is that if $U$ is not ...
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Prove $D(a;R_1,R_2)$ is a connected set

I was solving problems from the start of my Complex Analysis course, and I found this one (the beggining of my course focuses a lot in topology): Prove that $D(a; R_1,R_2)$ is a connected set. The ...
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Function connected implies Darboux function

Definition Let $f: X \rightarrow Y$ a function, $X,Y$ topological spaces. We say that $f$ is connected if $G_{\text{f}}(f)= \{(x, f(x)): x \in X\}$ is connected and we say $f$ is Darboux if $ \forall ...
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A region is simply connected if $\theta < 2\pi$, and not simply connected if $\theta \ge 2\pi$

The question is: Which of the following regions are simply-connected? (...) e) in polar coordinates, the region where $r>0,0<\theta<\theta_0$. The solution says: [simply connected] if $\...
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Image of an intersection is the intersection of the image [duplicate]

I'm studying the proof that path-connectedness implies connectedness. However there is one assertion that I don't understand. Let $f:[0,1] \rightarrow X$ be a continuous mapping where $X$ is a metric ...
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1answer
28 views

Given a continuous monotone function $f$, is $f^{-1}(x)$ a connected set?

Given a continuous monotone function $f: \mathbb{R} \rightarrow \mathbb{R}$, is it true that for any point $x \in \mathbb{R}$, $f^{-1}(x)$ must also be connected? The monotonicity is defined as a non ...
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1answer
29 views

Show that the set $S = \{(x_1, x_2, x_3) : x_1^2 + x_2^2 = x_3^2, (x_1, x_2, x_3) + (0,0,0)\}$ is not connected

This problem is in my text: In $R^3$ with the Euclidean metric, show that the set $S = \{(x_1, x_2, x_3) : x_1^2 + x_2^2 = x_3^2, (x_1, x_2, x_3) + (0,0,0)\}$ is not connected. I'm stuck on the $(...
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Is every open and connected set in $\mathbb C$ the continuous image of the open unit disk?

Let $$ \mathbb D=\{z\in\mathbb C\ |\ |z|<1\} $$ be the open unit disk in $\mathbb C$. It is well known that an open (nonempty) set $U\subseteq\mathbb C$ is simply connected if and only if it is ...
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Prove that $S^3\setminus S^1$ is connected

As the titles says, I have to prove that $S^3\setminus S^1$ is connected. I'm having a hard time solving this problem. The best idea I've come up with is to show somehow that, if $S^3\setminus S^1$ ...
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1answer
51 views

Let $\mathcal{f}: \mathbb{R} \to \mathbb{R}$ be a continuous function and consider $\mathcal{F}=[(x,y)\in \mathbb{R}^2 \mid y>f(x)] $.

Let $\mathcal{f}: \mathbb{R} \to \mathbb{R}$ be a continuous function and consider $\mathcal{F}=\lbrace (x,y)\in \mathbb{R}^2 \mid y>f(x) \rbrace $.Prove that $\mathcal{F}$ is path-connected. ...
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2answers
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Whether the subspaces are homeomorphic or not

From topology without tears, Let $X$ be a unit circle in $\mathbb{R}^2$,that is , $X=\{(x,y):x^2+y^2=1\}$ and has subspace topology. $Y$ be a subspace in $\mathbb{R}^2$ given by $Y=\{(x,y):x^2+y^2=...
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1answer
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Prove that $X$ is connected

Let: $$S=\{(p,q,0)\in \mathbb{R}^3: p^2 +q^2<1, p\in \mathbb{Q}, q\in \mathbb{Q} \}$$ $$I(a,b) \text{ - open line segment between points } a, b$$ Now let's define set $X$: $$X=\{(x,y,0) \in \mathbb{...
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Connectedness between subsets

I know a theorem: If in a topological space $(X,\tau)$ we have: $S\subset T\subset \overline{S}$ and $S$ is connected set then also $T$ is connected. In particular, the closure of a connected space ...
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Is the local connectedness heritable over the connected subspaces?

Definition A topological space $X$ is locally connected if each point $x\in X$ has a base of connected neighborhoods. So let be $X$ a locally connected space and let be $Y$ a connected subspace. So if ...
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1answer
47 views

Set Y which is not pathwise connected.

In topological space $\mathbb{R}^2$, let $A=\{(x,y):y=\frac{x}{n}, n \in \mathbb{N}\}$ and $B=\{(1,0)\}$. Let $Y=A\cup B$, then how can I prove $Y$ is not pathwise connected. the definition of path ...
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1answer
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Prove that a real normed space $V$ of dimension $n$ is disconnected in two connected parts by any linear subspace of dimension $n-1$

Theorem All norms on a finite dimensional vector space $V$ are equivalent. Proof. Omitted. So to follow we will use the infty norm $\lVert\cdot\lVert_\infty$ given by the equation $$ \lVert\vec v\...
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54 views

A directed inverse limit of finite connected spaces is connected

Apparently, a directed inverse limit of finite connected spaces is connected. Does anyone have a reference?
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1answer
40 views

Some characterization of finite complement topology [closed]

Let $Y:=\{(u,v)\in \mathbb{R}^2: u^2+v^2\le r\}$ where $r>0$ is finite complement topology. Now, If I want to characterize all the connected subsets of $Y$, then how should I start for such ...
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Prove the following functions is a Darboux function but discontinuous everywhere

Definition Let $X$ and $Y$ be topological spaces. We say $f: X \rightarrow Y$ is Darboux provided that $F[C]$ is connected for every connected C $\subset$ X. Problem Consider the Cantor set $C$ in $[0,...
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1answer
36 views

On a problem of connectedness in $\mathbb{R}^{n}$

I'm trying to solve a problem from the book "Foundations of real and abstract analysis" by Douglas Bridges. I'm asking here because I think a have a solution to the excersice but I'm not ...
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1answer
57 views

Boundaries in Spaces where Quasicomponents and Components Coincide

Let's call a space $X$ geometric if its components and quasi-components coincide. Let's also define a property called the boundary bumping property: $X$ has the boundary bumping property ("bbp&...
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Connectedness condition by basis\sub-basis

I know that some basic topological properties like compactness, $T_0$ and $T_1$ are among the properties which can be tested by a basis of the topology. I was wondering whether there is such a ...
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52 views

Let $X\subset \mathbb{R^2}$ be bounded and convex.Show that $\mathbb{R^2} \setminus X$ is path connected

Let $X\subset \mathbb{R^2}$ be bounded and convex.Show that $\mathbb{R^2} \setminus X$ is path connected. I understand the proposition, I was trying proof by contradiction,but I don´t get inspired, ...
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Duality Between Semilattices and Totally Disconnected locally Compact Hausdorff Spaces

On page 18 of this paper, the author states that there is a duality (correspondence?) between semilattices (i.e., abelian semigroups of idempotents) and totally disconnected locally compact Hausdorff ...
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2answers
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Theorem about connectedness.

$(X, \mathcal{O})$ : topological space $\{ M_{\lambda} | \lambda \in \Lambda \}$ : subset system of X Theorem; \begin{equation} \forall \lambda \in \Lambda ; M_{\lambda} \text{ is connected space}, x \...
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1answer
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If $X$ is locally path connected then the components and the path components are the same

Hi i am reading Topology by Munkres and there in theorem 25.5 i have two doubts.In the proof it says that Since $P$ is connected, $P\subset C$. Then at another line it says each of them necessarily ...
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4answers
54 views

Show that $T$ contains at most two points

Take $n \geq 2$ and $f:S^{n} \to \mathbb{R}$ a continuous map.Call $T$ the set of points $t\in f(S^{n})$ such that the fibre $f^{-1}(t)$ has finite cardinality. Show that $T$ contains at most two ...
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1answer
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Components of $X$ are connected disjoint subspaces whose union is $X$ s.t each nonempty connected subspace intersects only one of them

Hi i am reading Topology by Munkres and have one doubt in the proof of theorem 25.1. There it says By the result just proved $A_x\subset C$ I am unable to understand the above quoted statement. The ...
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1answer
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removing a point from a unit circle results in a connected space

This is an example our TA gave us. "If we removed a finite numbers of points of the unit circle the resulting set is connected" Can somebody explain why??
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Prove that there exists at most one vector $v$ orthogonal to $p-q$ and such that $x$ belongs to the range of $\alpha_{p,q,v}$

For any triple $p,q,v \in \mathbb{R^n}$ the path $$\alpha_{p,q,v}:[0,1] \to \mathbb{R^n},\alpha_{p,q,v}(t)=(1-t)p+tq+t(1-t)v$$ Is a parabolic arc with endpoints $p,q$. Prove that if $p\neq q$, for any ...
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1answer
31 views

Let $A$ be a connected subspace of $X$.If $A\subset B\subset \bar A$,then $B$ is also connected.

Hi i am reading Topology by Munkres and i have 2 doubt in the proof of theorem 23.4.The part that i have problem in is: $B$ cannot intersect D. This contradicts the fact $D$ is a nonempty subset of $...
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1answer
33 views

Where does surjectivity fail in a mapping from the unit interval to the Cantor Set?

I'm asked to state why there cannot be a surjective and continuous function $$g: [0,1]\longrightarrow \text{Cantor Set} $$ I know that $g^{-1}$ exists and is continuous & surjective since the ...
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1answer
23 views

Connected Subspace lies entrirely in $X$ [duplicate]

Hi i am reading Topology by Munkres and i have one doubt at the last line of lemma 23.2. It is written that Hence Y must lie entirely in C or in D. I can't understand the above quoted statement. ...
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Prove that if $X \subseteq \mathbb{R}$ is an interval then $X$ is connected.

Prove that if $X \subseteq \mathbb{R}$ is an interval then $X$ is connected. I found this proof in a book but I really don't understand it, and I would like to know if someone could please explain. ...
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1answer
75 views

Proving that the complementary of a surface is connected

I must prove that, being $S$ a differentiable surface of dimension $d$, which is a closed subset of $\mathbb{R}^k$, where $k\geq d+2$, one have $\mathbb{R}^k-S$ connected. This is trivial for $d=1$, ...
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1answer
26 views

Is connectedness relative property

I am bit confused after reading definition 2.45 from baby Rudin. 2.45   Definition   Two subsets A and B of a metric space X are said to be separated if both A∩B¯ and A¯∩B are empty, i.e., if no ...
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1answer
22 views

How can I tell if a set is compact and connected?

For expample if I consider the subspace $A_n$ of $\mathbb{R}^2$ defined by $$A_n=\{(x,y)\in\mathbb{R}^2:y=|x|^{2n+1}\}$$ and $$X=\bigcup_{n\ge0} A_n$$ Using Heine-Borel I just have to prove it's ...
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1answer
12 views

Constructing a continuous simple approximation of the identity map for a totally disconnected compact metric space

In Petersen's Ergodic Theory, when talking about totally disonnected spaces (Section 4.4), he says that if $D$ is a compact, totally disconnected subspace of $\mathbb{R}$, then for every $\delta > ...
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60 views

A snow of connected components

In short: snowflakes look like small connected components of random graphs; are there models using this, and what kind of model would make sense, if any? While I was once again fascinated by falling ...
2
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1answer
26 views

Is it true that if the property “being a closure of its interior” holds for a set, then it also holds for each connected component of that set?

Suppose that $A\subseteq\mathbb{R}^k$ is the closure of its interior. Is it true, then, that each connected component of $A$ is the closure of its respective interior? I.e., does it hold that $$\tag{1}...
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221 views

Existence of a path between two sides of a square, which is contained in the intersection of two open subsets of it

Denote by $I$ the interval $[0,1]$. Let $A,B$ be two open subsets of the square $I\times I=I^2$ such that $A\cup B=I^2$, $\{0\}\times I\subseteq A$ and $\{1\}\times I\subseteq B$. In more geometric ...

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