Questions tagged [connectedness]

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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1answer
233 views

Equivalence of Path-Connectedness and Arc-Connectedness for Hausdorff Spaces

I have a classical sort of question. If we define a path in a space $X$ from points $a$ to $b$ to be a continuous image $f: I \rightarrow X$ from the unit interval to $X$ such that $f(0) = a$ and $f(...
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0answers
15 views

Divide a grid into K connected components using linear programming [on hold]

I am given a $N \times M$ grid and I want to divide it's cells into $K$ connected components and I also want to know which component a cell belongs to.
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32 views

Problem with connectdness and and the diffrential of function.

I was reading lecture notes of Werner Ballmann: Automorphism groups here, and get stuck in the following two Lemma's 2.4 and 2.5, more precisely in lemma 2.4 I did try using flow properties, ie. $\...
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1answer
42 views
+50

Counting tranversals in an intersecting family with bounded number of connected components

Let $X$ be a set and $\mathcal{F}$ be a finite family of subsets of $X$. We call a set $T\subseteq X$ a tranversal set for $\mathcal{F}$ if it intersects every set in $\mathcal{F}$. Suppose that $X=\...
12
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2answers
165 views

Connectedness and path connectedness of a set whose intersection with lines is open

Let $A \subseteq \mathbb{R^2}$ a not open set with the property: Its intersection with every line L is open in L with the induced Euclidean topology. If the set is connected is it necessarily path ...
0
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1answer
10 views

Deleted comb space modified to be locally connected at zero

Let $D$ refer to the deleted comb space. I read that if we add to $D$ all points of the form $\{0\} \times \{1/n\}$ for $n \in \mathbb{Z}_+$, the origin becomes locally connected but not locally path ...
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2answers
413 views

Find a topological space $X$ which is connected but has three path components.

What would a connected topological space $X$ look like with three path components? I know that since it has a finite number of path components, these components are closed but I'm not sure if that ...
7
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2answers
671 views

Examples of extremally disconnected spaces

I am trying to understand the notion of extremally disconnected space (in other words Stonean space), i.e. a space in which any open set has an open closure. Could you help me and give (reasonable) ...
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2answers
88 views

How to find continuous functions that demonstrate that the set $\{(x,y):y>x\}$ is open and connected?

Consider the set of points $U$ in $\Bbb{R}^2$ that lie above the line $y = x$, i.e. points $(a,b)$ such that $b>a$. Prove that $U$ is open and connected. The method that is recommended is showing ...
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1answer
41 views

Connected Sets in the Complex Plane

I am trying to solve the following problem in Brown and Churchill's Complex Variables text. Let $S$ be the open set consisting of all points $z$ such that $|z| < 1$ or $|z - 2| < 1$. State ...
2
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1answer
61 views

Path connectedness in unit disk

Let $D\subset \mathbb{R}^2$ be the closed unit disk, boundary of $D$ is $\mathbb{S}^1$, let $a,b\in \mathbb{S}^1$ are two distinct points, $A,B\subset D$ are two disjoint closed sets with $A\cap \...
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2answers
784 views

Product of connected sets is connected

I know that this question had already been asked here but there is a problem ... all the proofs that I've seen used homeomorphisms and continuous functions. Well my teacher didn't teach me what a ...
1
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1answer
44 views

When does path connectedness implies convexity?

It is easy to see that every convex set is path connected. What are some examples so that converse holds (not counting the (trivial) one dimensional case)? Is there a nice topology so that this holds? ...
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0answers
43 views

equivalence of arakelian set definition

A closed set $E$ without holes ( holes means bounded component of its complement) is said to be Arakelian if for every disc $D$ ( or compact sets in general) the union of all holes of $E \cup D $ ...
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1answer
18 views

A $2$-dim manifold $M$ is contained in a sphere around the origin if and only if every $\gamma :[0,1] \to M$ satisfies $\int_\gamma xdx+ydy+zdz = 0$

Let $M \subset \mathbb{R}^3$ be a $2$-dimensional smooth and connected manifold. I need to prove that the following two claims are equal: For every smooth path $\gamma :[0,1] \to M$ it satisfies $$\...
44
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4answers
2k views

Can a nowhere continuous function have a connected graph?

After noticing that function $f: \mathbb R\rightarrow \mathbb R $ $$ f(x) = \left\{\begin{array}{l} \sin\frac{1}{x} & \text{for }x\neq 0 \\ 0 &\text{for }x=0 \end{array}\right. $$ has a graph ...
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3answers
27 views

If every Component in a compact space is open then the number of components is finite.

Let $X$ a compact set. Prove that if every connected component is open then the number of components is finite. Ok, $X = \bigcup C(x)$ where $C(x)$ is the connected component of $x \in X.$ I know ...
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1answer
23 views

Connected spaces and constant function

Let $Y$ a discrete space. Prove that a space $X$ is connected if only if every $f:X\to Y$ is constant. My incomplete attempt: Ok, If $X$ isn't connected then there are $A, B $ open sets such that $...
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4answers
30 views

Related to Compact and Connected set

Is the following set $S$ compact subset of $\mathbb{R}^2$? $$S= \{(x,y) \in \mathbb{R}^2 | xy<0 \}$$ Is the set $S$ connected subset of $\mathbb{R}^2$? I've done the compact part by looking ...
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1answer
19 views

Connected sets and irrational numbers [duplicate]

Prove that the set $C$ of points in $\mathbb{R}^2$ such that at least a coordinate is irrational is connected set. Ok, I know that the set of Irrational numbers isn't connected but how that helps me? ...
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2answers
21 views

Components of any set X

Given X, define an equivalence relation on X by setting x~y if there is a connected subspace of X containing both x and y. The equivalence classes are called yhe components of X. This is what I ...
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1answer
44 views

Topological spaces and Path-connected space

Let $\{A_i\}$ a family of connected subsets on topological space $X$ and let $A_0 \in \{A_i\}$ such that $A_i \cap A_0 \neq \emptyset.$ Prove that $\bigcup A_i$ is connected. My Attempt: Call $S= \...
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0answers
83 views

A set which is the closure of its interior points

I am trying to give a sufficient condition for a set in $\mathbb{R}^n$ which is the closure of its interior points. A priori, such a set has to be a closed set. A closed set in general is not the ...
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0answers
20 views

Bridgeless cubic graph has a 1-factor not containing two arbitrarily prescribed lines

According to Petersen's theorem, every bridgeless cubic graph has a perfect matching. While studying the proof of Petersen's theorem I came accross the following theorem "every bridgeless cubic graph ...
6
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1answer
1k views

If $p : E \to B$ is a covering map and $U \subset B$ is connected and evenly covered, then $p^{-1}(U)$ as a partition into slices is unique.

If I have a covering map $p:E \rightarrow B$ and some connected set $U$, that is evenly covered, then $p^{-1}(U)$ as a partition into slices is unique. Now, if I assume that $B$ is connected, then I ...
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1answer
241 views

If $Y$ is simply connected, then it doesn't admit covering maps that aren't homeomorphisms

Let be $Y$ a simply connected space. Show that $Y$ doesn't admit covering maps that aren't homeomorphisms, ie, every cover space of $Y$ is trivial ($I\times Y$, with $I$ a discrete space). So, I know ...
2
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1answer
276 views

If $X$ is a covering space of $Y$ with the covering map $p$ and $Y$ is connected, then $p^{-1}(y)$ have the same cardinality for every $y\in Y$. [duplicate]

I need to show that if $X$ is a covering space of $Y$ with the covering map $p$ and $Y$ is connected, then $p^{-1}(y)$ have the same cardinality for every $y\in Y$. I have this hint: A function $f:W\...
2
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3answers
636 views

If $h : Y \to X$ is a covering map and $Y$ is connected, then the cardinality of the fiber $h^{-1}(x)$ is independent of $x \in X$.

In "Knots and Primes: An Introduction to Arithmetic Topology", the author uses the following proposition Let $h: Y \to X$ be a covering. For any path $\gamma : [0,1] \to X$ and any $y \in h^{-1}(x) (...
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2answers
412 views

If $p : E \to B$ is a covering map and $E$ is simply connected, then each fiber of $p$ has the cardinality of that of $\pi_1(B)$.

Can anyone provide a source or a proof of the following fact: If $p : E \to B$ is a covering map and $E$ is simply connected, then each fiber of $p$ has the cardinality of that of $\pi_1(B)$?
3
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1answer
628 views

If $p : E \to X$ is a covering map and $E$ is path connected, then all the fibers have the same cardinality. [duplicate]

Let $p:E \rightarrow X$ be a covering space. It is well known that if $X$ is connected, then all the fibers have the same cardinality. This can be seen as a simple consequence of the fact that the ...
2
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0answers
24 views

$n$-connected region, confusion with definition

A region $G \subset \mathbb{C}$ is called $n$-connected if $\mathbb{C}_{\infty} \backslash G$ has $n+1$ components (Conway chapter 15). Consider now two regions: an annulus $A=\{r<|z|<1\}$ and ...
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0answers
17 views

Proof space is Connected Space [duplicate]

Let $A:=\{(0,y):y\in[-1,1]\}$ and $B:=\{(x, sin({1\over x}):x \in (0,1]\}$, finally, let $C:=A\cup B$ Show C is a connected space Show C is not path-connected 1 has me stumped, I have only ever ...
2
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3answers
45 views

Cardinality of the family of all possible connected subsets of $\Bbb R^n$

Actually, I have tried the obvious fact that a subset of the line is connected iff it's an interval. And, the family of all possible intervals of the line is equinumerous with $\Bbb R$, as we can send ...
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1answer
29 views

Definition of disconnected set in topology

I'm reading through wikipedia for a rigorous definition of disconnected topological space, which is the same as the one given by Munkres. A topological space $X$ is said to be disconnected if it is ...
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0answers
25 views

Characteristic of a connected, locally compact, Hausdorff space X which is locally connected.

Q. Prove that a connected, locally compact, Hausdorff space X is locally connected if and only if for each compact subset K and each open set U containing K, all but a finite number of components of X-...
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0answers
56 views

Compact + chain-connected metric space => connected

How to show that if $M$ (a metric space) is compact and chain-connected then it is connected ? Definition of $\varepsilon$-chainable : $(X,d)$ is $\varepsilon$-chainable if given any two points $a,b\...
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1answer
40 views

Boundaries of connected components

Let $ K \subset \mathbb{R}^{n}$ a compact set and $L_i$ the connected components of $\mathbb{R}^{n} \backslash K$. I can't see why $\partial L_i \subset K$.
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15 views

Connected Cauchy-hypersurface

Let $(M,g)$ be a connected globally hyperbolic spacetime with noncompact Cauchy hypersurface $S$. Can we always follow, that $S$ is also connected?
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1answer
23 views

Is $\overline{B_1((1,0))}\cup B_1((-1,0))$ connected? Is it path connected?

I'm trying to solve a question which asks me to let $B_1(p)$ denote the unit ball around $p$ in $\mathbb{R}^2$. I'm supposed to decide whether $\overline{B_1((1,0))}\cup B_1((-1,0))$ is connected and/...
2
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2answers
173 views

minimal embeddings of topological spaces into connected spaces

Defintions: Let $X$ be a topological space. 1) A connected space $Y$ is a minimal connected ambient (m.c.a for short) space for $X$ if there exists an embedding $i:X\mapsto Y$, and for every ...
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2answers
46 views

Connected and path-connected sets

I am struggling to work out a way of determining whether a set is connected or not. I have 3 examples I am looking it, and I am struggling to figure out which of them are connected/path-connected. $...
0
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1answer
56 views

$B=\{x \in A: d(x,\partial A)\geq \varepsilon\}$ is connected

Let $A$ be a compact and connected subset of $\Omega$, where $\Omega$ is a open subset of $\mathbb{R}^{n}$. I would like to know if there exists a $\varepsilon>0$ such that $B=\{x \in A: d(x,\...
1
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1answer
28 views

Why is the deleted comb space still connected?

I read from wikipedia that the deleted comb space: $$ (\{0\} \times \{0,1\}) \bigcup (K \times [0,1]) \bigcup ([0,1] \times \{ 0 \}) $$ has a well known result of being connected.. But how is it ...
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0answers
60 views

Connected components of an open subset

Suppose we have an (affine) irreducible algebraic variety $X$ over $\mathbb{R}$, and we are given one polynomial $f$. Consider the closed subset $C$ where this is zero. What can we say about the ...
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0answers
36 views

Prob. 6, Sec. 27, in Munkres' TOPOLOGY, 2nd ed: The Cantor's set is totally disconnected, compact, and uncountable

Here is Prob. 6, Sec. 27, in the book Topology by James R. Munkres, 2nd edition: Let $A_0$ be the closed interval $[0, 1]$ in $\mathbb{R}$. Let $A_1$ be the set obtained from $A_0$ by deleting its "...
3
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1answer
76 views

Probs. 3 (a & c), Sec. 27, in Munkres' TOPOLOGY, 2nd ed: The space $\mathbb{R}_K$ is not path connected and $[0, 1]$ is not compact in $\mathbb{R}_K$.

Here is Prob. 3, Sec. 27, in the book Topology by James R. Munkres, 2nd edition: Recall that $\mathbb{R}_K$ denotes $\mathbb{R}$ in the $K$-topology. (a) Show that $[0, 1]$ is not compact as a ...
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0answers
18 views

How is a $k$-form integrated over an oriented smooth $n$-manifold in the case it is connected?

I have seen in several answers to questions on this page stating that there is no way to integrate a $k$-form over an oriented smooth $n$-manifold if $k \neq n$. However I cite Tu in his book on ...
1
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1answer
44 views

Is it possible that there is a connected topological space without path-connected subspace?

Is it possible that there is a connected topological space without path-connected subspace? Furtherly ~ Is that any connected topological space $X$ always has dense path-connected subspace? Or Is ...
0
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1answer
47 views

Question about closed sets and connectedness

I have this theorem. let $A$ and $B$ two closed sets of a topological space $E$, such that $A\cap B$ and $A\cup B$ are connected, then $A$ and $B$ are connected. I want to prove it directly ...
3
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1answer
864 views

Show that a smooth map $F : M \rightarrow N$ has Constant rank if $F$ has a linear coordinate representation.

Suppose $F:M\to N$ is a smooth map, with $M,N$ smooth manifolds and $M$ connected. I want to show that if for each $p$ in $M$ there exists smooth charts near $p$ and $F(p)$ in which the coordinate ...