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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Connected set which is no-where path connected

Background: It's a fun exercise to try to construct a connected space $T$ such that no two points in $T$ can be connected with a path. My solution to the puzzle was to use an order topology on a ...
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Does there exists any locally compact haussdoff space $(X, \tau) $ which is totally disconnected but not zero dimensional?

• $(X, \tau)$ is zero dimensional if it has a clopen basis. • $(X, \tau)$ is totally disconnected if it all components has cardinality $1$. •$(X, \tau)$ is locally compact haussdoff space if $\forall ...
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An open, limited and connected set $C$ under the hipotesis of the theorem of inverse function with $f(\partial C)\cap C = \emptyset$

PROBLEM: Consider $(V,||·||_{V})$ Banach, $U\subset V$ open and $f:U\rightarrow V$ of class $C^{1}$ with Frechet derivative invertible in all $U$. Supose that $A\subset U$ is open, connected and $\bar{...
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Relation between connectedness and Dedekind Completeness

In my calc class we constructed the real numbers using the following 5 axioms: The set is nonempty. The set has an ordering. The set has no first/last point. The set is connected. The set contains a ...
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Boundary of an open bounded connected subset of $\mathbb{R}^2$ must contain the image of a path?

How does one prove that the boundary of a bounded open connected subset of $\mathbb{R}^2$ must contain the image of a path? I require a non-constant path. But it doesn't have to be injective. This ...
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1 answer
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What ensures the existences of such sequence

Let $(X,d)$ be a metric space and $A \subset X$ a connected subset. Let $B \subset X$ s.t. $A\subset B \subset \overline A$. Then $B$ is connected. I have a proof in my textbook that I understand ...
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Can quasi-concave and monotone functions level curves that are not path-connected?

For $X = \mathbb{R}^2$, does there exist a quasi-concave and monotone function $f : X \to \mathbb{R}$ that has a level curve which is not path-connected? Secondly, will every level curve necessarily ...
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2 answers
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Path connectedness equivalent to connected in euclidean topology for quasiprojective vareities?

Let $V$ be a quasiprojective real variety. My intuition tells me that this type of space with the Euclidean topology has the property that path-connectedness and connectedness are equivalent. Is this ...
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3 votes
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Understanding the following proof about path-disconnectedness of topologist sine curve.

Given the function $f$ from $\Bbb R^+_0$ to $\Bbb R$ defined by the equation $$ f(x):=\begin{cases}\sin\Big(\frac 1 x\Big),\,\text{if }x\in\Bbb R^+\\0,\,\text{if }x=0\end{cases} $$ for any $x\in\Bbb R^...
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Countable $T_1$ spaces are totally path disconnected

Steen & Seebach, Counterexamples in Topology, have the following as problem 30 on p. 206: Every countable $T_1$ space is totally path disconnected. I can see why a Hausdorff countable space $X$ ...
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Question on connected metric spaces and continous functions

This question came up in some sample questions my teacher sent for an upcoming test. I tried the part a with the theorem that the preimage of a non constant function would not be connected but I got ...
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Topological spaces whose quasi-components are connected

Let ${\mathcal X}$ be the category of topological spaces whose quasi-components are connected (and all continuous functions between them). I know that compact Hausdorff spaces are in ${\mathcal X}$. I ...
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Is the following converse of my first question true?

Is the following true? If C is a compact subset of an ordinary plane P, O a point of P , U a simply connected open set of the plane containing C but not O, then O belongs to the unbounded component of ...
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2 votes
1 answer
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Preimage of zero under a continuous function on compact real interval has at most countable connected components

As part of a larger inquiry, I suspect and am trying to prove the following : Let $\phi$ be a continuous function defined on $[0,1]$. Then $\phi^{-1}(0)$ has an at most countable number of connected ...
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Can any domain in the Riemann sphere be approximated by finitely connected Jordan domains?

Is it true that for any domain $\Omega \subset \bar{\mathbb{C}}$ we can find finitely connected Jordan domains $\Omega_n$ such that $$ \Omega_n \subset \bar{\Omega}_n \subset \Omega_{n+1} \subset \...
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Is a compact set C in the plane contained in an open simply connected set avoiding a point of the unbounded component of C's complement?

How does one prove that: If C is a compact subset of an ordinary plane P, O a point of P belonging to the unbounded component of P\C, then there exists a simply connected open set of the plane ...
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Characterizing functions with connected graph

The following characterization of functions $f:[a,b]\to\Bbb R$ with connected graph seems to be "well known", but I could not find a proof: Let $f:[a,b]\to\Bbb R$ be a function. The ...
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Global Minimizer.

I find the following thread If $\overline{x}$ is a point such that $f(x)=f(\overline{x}) \Rightarrow$ $x$ is a local minimizer, then $\overline{x}$ is global minimizer I am trying to understand the ...
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Let be $X$ a connected Linearly Ordered Topological Space: so if $Y$ is a nonempty bounded subset of $X$ then it has supremum or infimum.

Let be $(X,\preceq)$ a connected Linearly Ordered Topological Space. So I would like to prove or disprove the following statement $$ \text{Any upper/lower bounded nonempty subset $Y$ of $X$ has a ...
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2 votes
1 answer
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Proving that all fibers of a morphism $\varphi:X\to\Bbb{P}^1$ are connected

Let $X$ be a rational elliptic surface over an algebraically closed field $k$ and $\pi:X\to\Bbb{P}^1$ its elliptic fibration, which I assume is relatively minimal. If $D$ is a nef divisor such that $D^...
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1 vote
1 answer
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Connected and Locally compact

Im having some trouble with the following: Let $X=(0,1)$ and $T=\{(0,1-\frac{1}{n}|n \in \mathbb{N},n\geq 2\} \cup \{X,Ø\}$ I want to prove that (X,T) is: Connected The definition of being connected: ...
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If $S_i\cap S_{i+1}\ne \emptyset$ and $S_i$'s are connected sets, then $\bigcup_{i=1}^{n}S_i$ is also connected?

Is the following statement about connected sets in metric space true? Statement: Let $S_1,S_2,\cdots,S_n$ be connected sets such that $S_i\cap S_{i+1}\ne \emptyset$ for all $1\le i\le n-1$. Then $S=\...
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0 answers
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Union of simply-connected domains is simply-connected

Let $(D_n)_{n\in\mathbb{N}}$ be a sequence of simply-connected domains in $\mathbb{C}$ so that $D_n\subset D_{n+1}$. Prove: $D:=\bigcup_{n=1}^{\infty}D_n$ is a simply-connected domain. I have already ...
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-1 votes
1 answer
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Is this set connected or path connected? [closed]

How to check the $S =\{(x,y)\mid x+y \in \mathbb{Q}\} \subseteq \mathbb{R^2}$ under usual topology is connected or path-connected? My approach is, not path-connected because if we pick any two points ...
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0 votes
1 answer
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Preimage of simply connected set

Let $G\subset\mathbb{C}$ be a domain and $f:G\to\mathbb{C}$ a holomorphic, non-constant function. Prove or disprove: If $f(G)$ is simply connected, then $G$ is simply connected. My idea: I know that ...
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14 votes
1 answer
114 views

Can we describe the connected components in the graph of $\cos\left(x\right)-\sin\left(x+y\right)=\cos\left(x^{2}y\right)$?

The graph of $\cos\left(x\right)-\sin\left(x+y\right)=\cos\left(x^{2}y\right)$ is, perhaps unsurprisingly, pretty wild. Here is a Desmos version of it; a few screenshots are below, showing different ...
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Show that $M \setminus f^{-1}(c)$ is disconnected if $f:M\rightarrow \mathbb{R}$ is continuous and $c \in (\min f(M), \max f(M))$.

Let $f : M \rightarrow \mathbb{R}$ be a continuous function. Prove that if $c \in \mathbb{R}$ is strictly between minimum and maximum of $f$ in $M$ then $M \setminus f^{-1}(c)$ is disconnected. Can ...
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2 votes
1 answer
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Is there a nontrivial LOTS that is connected and totally path disconnected?

By nontrivial LOTS I mean a linearly ordered space that contains more then one point. Being totally path disconnected means that every path in the space is constant. A connected linearly ordered ...
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1 vote
0 answers
26 views

Connected complement of compact subset

Let $U$ be an open subset of $\mathbb{R}^n$ and $K$ a compact subset of $U$ such that $U\setminus K$ is connected. Does there exist an open set $V$ such that $K\subseteq V \subseteq \overline{V} \...
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0 answers
61 views

Closed-and-open sets and connectedness

Let $f: [0,1] \to [0,1]$ and $g: [0,1]^2 \to [0,1]$ be continuous with $f(0) \leq g(0,t)$ and $f(1) \geq g(1,t)$ for all $t \in [0,1]$. Let $$X = \{(x,t) \in [0,1]^2: f(x) = g(x,t)\}.$$ Question: Can $...
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2 votes
1 answer
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Computing $\pi_1$ of subset $X\subseteq \mathbb R^2$, $X$ union of 3 simp.conn. subspaces w/ simp.conn. pairwise intersection but empty intersection

Let $X$ be a subset of $\mathbb R^2$, and suppose that $X$ is equal to the union of open and simply connected subspaces $V_1,V_2,V_3$. Moreover, assume that the pairwise intersections $V_i \cap V_j, \...
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3 votes
1 answer
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Is the space of $n\times n$ real symmetric matrices with strictly positive determinant connected within the vector space of $n\times n$ real matrices?

I want to make clear that I am aware of the connectedness in the case of general real matrices. But here I ask about the subspace of symmetric ones. If it is not the case, which are the connected ...
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1 answer
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Show that X is connected if and only if: $\forall O_1,O_2 \in \tau: X=O_1\cup O_2 \land O_1\cap O_2=∅ \Rightarrow O_1=∅ \lor O_2=∅$

Let X be a nonempty set with topology $\tau$. Show that X is connected if and only if the following implication holds: $\forall O_1,O_2 \in \tau: X=O_1\cup O_2 \land O_1\cap O_2=∅ \Rightarrow O_1=∅ \...
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1 vote
1 answer
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Show that $\{(x,y): 0 < x \leq 1; y = \sin{1/x}\} \cup \{z: x = 0; -1 \leq y \leq 1\}$ is not path connected [duplicate]

Show that $\{(x,y): 0 < x \leq 1; y = \sin{1/x}\} \cup \{z: x = 0; -1 \leq y \leq 1\}$ is not path connected. The book gives this set as an example of a closed, connected but not path connected set....
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-1 votes
1 answer
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Question on the connectedness of the orthogonal group

I want to show that the quotient $O_2^- = O_2/SO_2$ is connected. My idea was as follows: It's easy to show that $SO_2$ is connected. $S0_2$ is a topological group (normal subgroup of a topological ...
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1 vote
0 answers
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Union of a connected subset with a component is connected

If $X$ is a connected topological space, and $Y$ is a connected subspace, then for any connected component $C$ of $X\backslash Y$, $Y \cup C$ is connected. Results proven before: Connected components ...
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1 answer
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Can a simply-connected topological space be written as the union of the images of all possible loops at a given basepoint?

Can a simply-connected topological space be written as the union of the images of all possible loops at a given basepoint? Let $X$ be a simply-connected topological space, fix $x\in X$, and let $\{\...
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1 answer
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If $X$ has finitely many connected components then any connected component is open. Which subset in $X$ are open and closed?

I am currently reading Topology and for guidance I'm following my college lectures and the book by Munkres. I'm still in the beginning so I'm having a difficulty to understand the following : If $X$ ...
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1 answer
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Homological vs. homotopical connectivity

There are two closely-related measures of connectivity of a topological space: Homotopical connectivity is the largest $n$ such that all homotopy groups up to and including the $n$-th one are trivial;...
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0 votes
1 answer
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Find a continuous map to show a set in $R^2$ is connected

I want to show that the set $S = \{(x,y) \in \mathbb{R}^2: 0 \leq x \leq y^2, 1 \leq y \leq 2 \}$ is connected My first take is trying to show that $S$ is the image of a continuous function $f$ ...
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0 votes
2 answers
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Exercise 6, Section 23 of Munkres’ Topology

Let $A\subset X$. Show that if $C$ is a connected subspace of $X$ that intersects both $A$ and $X-A$ , then $C$ intersects $\operatorname{Bd}A$. My attempt: Approach(1): Assume towards contradiction, ...
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1 vote
1 answer
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Let $S$ be a connected surface where the curvature of all its geodesic curves is constant, $K=1$. Prove that $S$ is contained in a sphere

Let we have $S$, a connected surface. Let's suppose that the curvature of all its geodesic curves is constant, $K=1$. I have to prove that $S \subseteq \mathbb{S}^2$. $\mathbb{S}^2=\{(x,y,z)\in \...
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Prove uniqueness via a connectedness argument

I am reading an article about uniqueness of some evolution PDE and at some point, the author says that to show uniqueness on an interval $[0,T]$, by a "classical connectedness argument" (his ...
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  • 2,534
3 votes
1 answer
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Disconnected image of a derivative

I would like to know if there exists a differentiable vector-valued function $f:[a,b]\rightarrow \Bbb R^2$ such that the image $f'([a,b])$ of its derivative is disconnected. My Attempt First of all I ...
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2 votes
1 answer
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Counterintuitive topological result concerning connectedness

Problem If we consider $A,B,C,D$ to be the corners of the square $[0,1]^2$ where $A$ and $C$ are diagonally opposite (and so are $B$ and $D$). Then we can define $X$ to be a connected subset of the ...
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1 answer
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Counterexample of a disconnected topological space with open components but not locally connected

Theorem 25.3 in Munkres proved that a space is locally connected if the components of every open subset is open. What obviously follows from the theorem is that the components of a locally connected ...
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4 votes
2 answers
170 views

Intuitively, what is the difference between a "simply connected" set and a "locally connected" set?

I was looking into the Mandelbrot Set and saw a note that said it has been proven that the Mandelbrot Set is "simply connected" but it is still an open question of whether or not it is "...
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1 vote
1 answer
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Prove or disprove that a Jordan curve exists that contains two points.

Question (not homework): Let $\Omega$ be open and connected. Prove or disprove that if $z_1, z_2\in\Omega$ are distinct, then there exists a Jordan curve $\gamma\subseteq\Omega$ such that $z_1, z_2\...
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1 vote
0 answers
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Is shape morphism invariant to the cardinality of the set of components/quasicomponents?

Let $X$ and $Y$ be two topological spaces. We denote with $CX$ and $CY$ the sets of connected components and $QX$ and $QY$ the sets of quasicomponents for the corresponding spaces. It is well known ...
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  • 3,076
1 vote
1 answer
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Connected Sets after applying a continuous function

Prove that if $f: X \rightarrow Y$ is a continuous function of metric spaces and $X$ is connected, then $f(X)$ is also connected. I have tried to prove this by the contrapositive assuming $f(X) = A \...
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