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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8 views

possible embeddings for a $2$-connected planar graph

When I asked the question "cycles and faces in planar graphs", I learned that the numbers of vertices in the faces are not unique, if the planar graph is only $2$-connected. My question now is : How ...
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1answer
26 views

The family of open intervals that do not contain $0$

Let $T$ be the collection of all open sets in $\mathbb{R}$ not containing $0$ union $\mathbb{R}$ i.e $$T=\{(a,b)\subset\mathbb{\bar R}:0\notin(a,b)\}\cup\{\mathbb{R}\}$$ Then what is true about $T$? ...
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1answer
19 views

$3$-connected non-hamiltonian graph with at most $3$ independent vertices

Is there a $3$-connected non-hamiltonian graph with at most $3$ independent vertices ? I checked the graphs upto $9$ vertices and the cubic graphs upto $18$ vertices and did not find such a graph. ...
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23 views

Is every connected subset of the Sierpiński triangle arcwise connected?

I think this should be true. If it's indeed the case, it seems like this should be a known result, so references are welcome. I managed to prove that (assuming $S$ is the connected subset) $S$ ...
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33 views

Find a line with measure 0

A finite measure $m$ is defined on a $k$-connected set $D$, with $k>1$. You want to convert $D$ into a $(k-1)$-connected set without hurting the measure. Formally: prove that there is a set ...
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1answer
37 views

is the lexicographic order topology on the unit square connected/path connected?

I was wondering, given the lexicographic order topology on $S=[0,1] \times [0,1]$, is it connected (and path connected)? I found a reference to Steen's and Seebach's Counterexamples in Topology, and ...
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1answer
29 views

Connected subsets problem

Is it True or False: If $A$ and $C$ are connected subsets of the metric space $X$, and if $A \subset B \subset C$ then $B$ is connected. If true, prove it's true. If false, give a counter-example to ...
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2answers
97 views

When is the free loop space simply connected?

I am not sure if there is an obvious answer to this, but this has been bothering me. Let $X$ be a topological space. When is the free loop space, $LX$, simply connected? Correct me if I'm wrong, but ...
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1answer
11 views

properties of connected set

If $C $ is a connected set in a metric space $X$ & $C$ intersects both $A$ and $X\cap A^c (A\subseteq X)$ then can it be concluded that $C\cap \delta A\neq\phi$ ?
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1answer
13 views

graph of connected set [closed]

Let $X,Y$ be connected metric spaces.Assume that $X$ is connected & that $f:X\rightarrow Y$ is continuous.Is $\{(x,y)\in X\times Y:y=f(x),x\in X\}$ connected in $X\times Y$(w.r.t. product metric)? ...
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1answer
39 views

Is $SO(2,\mathbb R)$ connected

Is $SO(2,\mathbb R)=\{ A\in O(2,\mathbb R): det A=1\}$ connected?Why?
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1answer
57 views

A very weird connected subset of $\mathbb R^2$

Is it possible to construct a connected subset of the plane with the property that removal of any single point makes it totally disconnected? Any answer is appreciated..Thanks!!
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2answers
36 views

If there exists a continuous non-constant map to the integers, then the space is not connected

Let X be a topological space. show that if there exists a continuous, non constant map from X to the integers with the discrete topology, then X is not connected. So I know that connected subspaces ...
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1answer
33 views

Möbius Transformation: Have they argumentet correctly?

There is an example in my book where they do something I am not sure why their argument work. First I will present the problem short in text, then I will present it again with pictures from the book: ...
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2answers
24 views

Cardinality of the set of irrational numbers which is connected subset of rational numbers

Let X be a connected subset of real numbers. If every elements of X is irrational then what is the cardinality of X? We know cardinality of irrational numbers is same as the cardinality of real ...
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1answer
53 views

What's an easy way to show that $GL(n,\mathbb C)$ is connected? [duplicate]

I think I've to show it's path connected, but can't figure out the path functions explicitly. Can anyone give these path maps?
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1answer
31 views

Are the spaces of real orthogonal, complex unitary, hermitian or symmetric matrices connected?

I want to know which of these are connected and which are not. I think I've to take some continuous map from the set of matrices to $\mathbb R$ or $\mathbb C$ and interpret these matrix sets as ...
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2answers
39 views

Topological space which is not locally connected

In class we defined a locally connected space as a space that has a basis consisting of connected sets. I don't quite understand what a space which is not locally connected would look like. At least ...
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2answers
55 views

How many connected graphs with $n$ nodes are there approximately?

OEIS states the the number of graphs with $0,1,2,...,19$ nodes is $$1, 1, 1, 2, 6, 21, 112, 853, 11117, 261080, 11716571, 1006700565, 164059830476,$$ $$50335907869219, 29003487462848061, ...
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2answers
47 views

Show that the set U is unbounded

I am working on a practice qualifier problem: Let $f : \mathbb{C} → \mathbb{C}$ be an entire function with $f(z) \ne 0$ for all $z ∈ \mathbb{C}$. Define U = {z ∈ C : |f(z)| < 1}. Show that all ...
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0answers
23 views

Showing that the topologist's sine curve is not path connected using an argument of sequences.

I am familiar with many proofs of the fact that the set (defined below) is not path connected. My favorite uses the fact that $[0,1]$ is compact and another good one uses the intermediate value ...
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2answers
34 views

If $f$ is a cut edge in $G-e$, is $e$ a cut edge in $G-f$?

It seems not for any general $G$. But if $G$ is 2-connected does the hypothesis hold?
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3answers
68 views

construct a path between (-1,0) and (0,2)

So we are given a region S which is above the x-axis and between the semicircle of radius 1 and 2 centred at the origin. we are asked to construct a path that connect the point (-1,0) and (0,2)..and ...
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2answers
44 views

topological properties of a given set

Let us consider the set $X=C[0,1]$ with its sup-norm topology. Let $S $ be the set of all elements $f$ of $X$ such that $\int_0^1 f(t) dt=0$. Is $S $ compact and connected? To show $S$ compact I have ...
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1answer
30 views

What characteristics of binary space are needed to prove that it is totally disconnected?

In trying to prove that binary space (a homeomorphic space to the better known Cantor Set) is totally disconnected, what traits of the space do I need? Is binary space point-wise open (that would ...
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2answers
19 views

is C[a,b] with 1-norm connected

If C[a,b] denotes the set of all real valued continuous functions over [a,b] is it connected w.r.t. the 1-norm ?1-norm of a function f is defined to be integration of f from a to b.
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3answers
44 views

Set $S$ which is path-connected, but $\overline{S}$ is not path-connected

Suppose the set $$S := \left\lbrace x+i \sin \left( \frac{1}{x} \right) \Bigg\vert x \in (0,1]\right\rbrace \subseteq \mathbb{C}. $$ I want to show that $S$ is path-connected but $\overline{S}$ is ...
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1answer
80 views

Is the following subset of a plane connected? (picture)

It's the union of a sequence of interlocked "chains" formed from closed semicircles. The chains can be seen having different shades of gray in the picture. There's no line in the middle, just the ...
2
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1answer
26 views

Boundary bumping theorem

Boundary bumping theorem: Let $X$ be a compact connected space, $A$ its closed proper subspace, $C$ a component of $A$, then $C ∩ ∂A ≠ ∅$. ($∂$ means boundary.) I wonder if the compacness assumption ...
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0answers
20 views

continuity of maps

Let $D$ denote the closed unit ball and let $E$ denote the set $E:=\{(x,y)\in R^2 :2x^2 + 3y^2\le1\}$ and let S be the unit circle. If $f:D\rightarrow S$ is continuous then does there exist $x\in S$ ...
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1answer
18 views

Proof Verification : If $S$ is a metric space and $U(x)$ is the component of $S$ containing $x$, then is $U(x)$ closed in $S$ .

If $S$ is a metric space and $U(x)$ is the component of $S$ containing $x$, then is $U(x)$ open in $S$? Attempt: $U(x)$ is the union of all connected subsets of $S$ containing $\{x\}$. Hence, $U(x)$ ...
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1answer
42 views

(Path)Connectedness of $w^{2} = \sin z$.

I recently began to study Riemann surfaces, and I got a problem while checking some examples in the book. It is easy to see, for example, that the subset $\{(z, w)\in \mathbb{C}^2\mid w^{2} = \sin ...
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3answers
32 views

If $ y \in [0,1] \times [0,1]$, is $[0,1] \times [0,1]-y$ connected?

$(a)$ If $ y \in [0,1] \times [0,1]$, then prove/disprove that $[0,1] \times [0,1]-\{y\}$ is connected. $(b)$ Hence, prove that a continuous, onto mapping $f :[0,1] \rightarrow [0,1] \times [0,1]$ ...
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1answer
38 views

Planar graphs and connectivity

How many edges must a planar graph with $n$ nodes have that it is sure that it is a) connected b) biconnected c) triconnected In particular, are all planar graphs with $n$ nodes and $3n-6$ edges ...
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2answers
38 views

If $E$ and $F$ are nonempty subsets of $M$ and if $E \cup F$ is connected, show that $\bar{E} \cap \bar{F} \neq \emptyset$

I tried to prove this by contradiction: So suppose $\bar{E} \cap \bar{F} = \emptyset$, that means $\bar{E}$ and $\bar{F}$ are disjoint. But I get stuck here... I have no idea what to do... Any ...
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3answers
53 views

Is the one point compactification of the set $\{xy = 0\}\in \mathbb R^2$ minus any point connected?

Call the subset of $\mathbb R^2$ given by $xy = 0$ (ie. the coordinate axis) $X$. Let the one point compactification of $X$ be given by $X^*$. In my mind, I have each axis headed to the ...
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1answer
26 views

Connectedness and locally path-connectedness

I'm studying algebraic topology and I have this doubt: $X$ is a topological space connected and locally path-connected; does this imply that $X$ is path-connected ? Why ? Meaning of locally ...
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1answer
49 views

$k$-connected graphs containing trees

I've encountered the following problem in the book "Graphs and Digraphs" and I'm not sure how to do it. Show that every $k$-connected graph contains any tree of order $k+1$ as a subgraph.
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1answer
48 views

What happens to the connectedness of $\mathbb R^2$ when countable many points are removed?

Does $\mathbb R^2$ remain connected when countably many points are removed? Does it remain path connected? This is not homework but is in response to working several problems where countable or ...
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118 views

Does there exist a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?

Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...
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1answer
24 views

Two intersecting disks with subspace topology in $\mathbb {R}^2$

Take two closed disks as subsets of $\Bbb R^2$ such that they intersect at exactly one point. Let $\Bbb R^2$ have the standard euclidean topology $\mathcal J_E$ and give the above set the subspace ...
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207 views

Is bijection mapping connected sets to connected homeomorphism?

If $f : \mathbb{R}^{n} \to \mathbb{R}^{n}$ is a bijection, mapping connected sets to connected, is $f$ necessarily a homeomorphism? The converse is true, a well known property of homeomorphisms. I ...
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1answer
21 views

Proving G is 3 edge connected if G is connected graph and for every edge $e$, there are cycles $C_1$ and $C_2$ s.t. $E(C_1) \bigcap E(C_2) = \{e\}$

(1) If G is connected graph, and for every edge $e$, there are cycles $C_1$ and $C_2$ such that $E(C_1) \bigcap E(C_2) = \{e\},$ then $G$ is 3 edge connected. I'm trying to figure out how to prove ...
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1answer
37 views

if $A\subseteq \mathbb R^n$ is connected then is $A´$ (derived set) connected?

are the following statements true? 1) if $A\subseteq \mathbb R^n$ is connected then is $A´$ connected? 2)if $A´$ is connected then is $A$ connected? I can´t find any counterexamples. Can you help ...
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14 views

Examples of monotone mappings?

I am looking for some interesting (non-trivial) examples of functions between normal spaces which are perfect and monotone, i.e., functions which are surjective and closed preimages of singletons ...
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27 views

Help with continuum theory

A continuum is a compact connected Hausdorff space (sometimes metric is included in the definition). I have yet to find any references that help me understand composants and components of a ...
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2answers
37 views

Let $f:[a,b]\to\mathbb R$ continuous. Prove that $G=${${(x,f(x): x\in [a,b]}$} (graph of $f$) is connected

Let $f:[a,b]\to\mathbb R$ continuous. Prove that $G=${${(x,f(x): x\in [a,b]}$} (graph of $f$) is connected Suppose $G$ is disconnected then $\exists A,B$ relatively open disjoint sets so that $A\neq ...
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29 views

About level curves of a continuous function in a real square, and connectivity

Assume f is a continuous function on the (unit) square in real plane. Name the edges N,S,E and W in the natural way. Assume f is >0 at W edge and <0 on E edge. Intuitively it is clear that there ...
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1answer
44 views

Connected Sets on Metric Spaces

I'm taking a first course in real analysis, and we're using Rudin's Principles of Mathematical Analysis as our main (only) book. In chapter two, Rudin discusses basic topology from the point of view ...
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tupules $(x,y)$ with at least one entry rational is connected in $R^2$

I have studied connectedness and came across a problem which goes like this.. all the tuples $(x,y)$ with at least one entry rational is connected in $\Bbb R^2$. I have tried to prove it by ...