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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3
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1answer
37 views

Simply connected and connected in complex analysis

Can some one please help me with this, why is third set in the picture not simply connected. The definition of simply connected (in space of complex numbers) is: A set is said to be simply ...
0
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2answers
56 views

How to find a continuous function that demonstrates that the set $\{(x,y):y>x\}$ is open?

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 ...
0
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1answer
18 views

Weakly Connected Graphs

How is the following graph a weakly connected graph?
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1answer
32 views

If two non-disjoint subsets are connected, why does their union have to be connected?

So X and Y are two sets such that their intersection is nonempty. I want to show that if X and Y are each connected, together their union is connected. I tried proving this by contraposition and I've ...
0
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1answer
31 views

Holomorphic function and nth derivative.

Let $K$ be a open connected subset of complex numbers and $f$ holomorphic on $K$. If $f=0$ on some open disc $D$ in $K$, then is it true that $n$th derivative of $f$ is $0$ for all points in $D$ ...
0
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3answers
63 views

How to show that a real continous function with image in the rationals is constant?

Can someone please explain to me how I am supposed to approach this question: If $f: [0,1] \to \mathbb{ R}$ is continuous, and has only rational values, then $f$ must be a constant.
4
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0answers
55 views

Conjecture in continuum theory: my proof attempt

Conjecture. Suppose $X$ is a normal connected space such that every nondegenerate closed subset of $X$ is disconnected. Then every proper subcontinuum of $\beta X$ has empty interior. proof attempt. ...
3
votes
1answer
54 views

Theorem 4.22 from baby Rudin: continuity and connectedness

I have some parts that I don't understand from the given proof. The theorem is: If $f$ is a continuous mapping of a metric space $X$ in to a metric space $Y$, and if $E$ is a connected subset of $X$, ...
3
votes
2answers
49 views

Is $[0,1]^2 \setminus \{(a,b)\}$ connected?

I am pretty sure that this set is in fact connected but I am struggling to see how to prove it, it is simple to see that $[0,1] \setminus \{x\}$ is disconnected but I can't see how to relate ...
0
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2answers
25 views

Are points in different connected components separated by open subsets?

Decide if the following statement is true or false: If $a,b \in M$ belong to different connected components, then there exists a disconnection $M = A \cup B$ (with $A$, $B$ open and disjoint), ...
2
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0answers
33 views

Knots which are composed of several strands

In a math textbook and this article in NRICH, some problems deal with a special kind of knots: those which are formed from several strands: The problems ask if a given knot can be formed from just ...
2
votes
2answers
26 views

Consider $X=C[0,1]$ with its usual sup-norm topology.Let $S=\{f\in X :\int _0^1f\neq 0\}$.Is the set connected?

Consider $X=C[0,1]$ with its usual sup-norm topology.Let $S=\{f\in X :\int _0^1f\neq 0\}$.Is the set connected? I tried to conclude from the path connectedness of $S$ .But $S$ is not path connected ...
0
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1answer
37 views

Components are clopen in a space with a finite number of connected components

I'm having trouble understanding why this fact is true. A lot of sites just assume it with out reason and it doesn't seem so direct to me. Anyways, here is the theorem: For any topological space ...
0
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3answers
57 views

Normal matrices connected? [closed]

Is the set of all normal matrices connected in $M_n(\mathbb{R})$, where the metric is the usual metric of $\mathbb{R}^{n^2}$? ($A$ is normal iff $AA^{t}=A^{t}A$.)
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3answers
42 views

Path connectedness of the set of points $(x,y)$ where $x$ is rational or $y$ is rational [duplicate]

Prove that $X=\{(x,y) :x\text{ is rational or }y\text{ is rational}\}$ is path connected. So for every $(x,y)$ in $X$, I need to find a continuous function $f$ on $[a,b]$ such that $f(a)=x$ and ...
0
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3answers
38 views

Determine whether the set $X=\{(a,b) : |b|>e^a \}\subset \mathbb R^2$ is connected

Determine whether the set (as a subspace of $\mathbb R^2$) is connected. $$X=\{(a,b) : |b|>e^a \}$$ Thoughts: Not sure how to go about this question. I suppose look for a partition. Anyone got ...
2
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0answers
39 views

Example of a series parallel graph with toughness greater than $\frac{4}{7}$

Can anyone lead me to an example of a "more than $\frac{4}{7}$ tough series parallel graph"? Graph toughness is defined as $T = \min \left\{\frac{|a|}{\omega{(G\backslash A)}}\right\}$ over all ...
0
votes
2answers
29 views

Showing a topological space covered by connected subspaces is connected

'Let $X$ be a topological space and let $(U_i)_{i \in I}$ be a cover of $X$ by connected subspaces $U_i$. Supposed for all $i,j \in I$ there exists some $n \geq 0$ and $k_0,...,k_n \in I$ such that ...
1
vote
1answer
31 views

A connected path between shapes

This is a follow-up to this question: A continuous path between shapes . Let $A$ and $B$ be two measureable, bounded, connected subsets of $\mathbb{R}^2$ such that $A\subseteq B$. Does there exist a ...
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0answers
26 views

Helping me my study of introduction to analysis

I am a math major student who started study math now In my university class , my professor proposed me a few question and I thoought several hours but I can`t write logically so i ask about question ...
0
votes
1answer
20 views

Proof-verification: Components are mapped to components by homeomorphism

Show that every homeomorphism of metric spaces maps connected components to connected components. I come up with a proof, but I did not include the fact that homeomorphism is bijective: Let ...
0
votes
1answer
44 views

For every connected space X and an open cover U, every two points has a simple chain containing them

I am trying to prove this theorem saying: " A space X is connected, if and only if for an open cover U of X, every two points in X has a chain between them". I cant prove only if part (a connected ...
2
votes
0answers
28 views

Question on connectedness and components

We know that any connected subset $C$ of $\mathbb{R}$ is an interval, so that if $C$ is bounded, then $C$ must be of one of the following 5 types: $(a,b),(a,b],[a,b),[a,b]$ with $a < b$, and $[a,a] ...
1
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2answers
86 views

Would a connected space contain a compact subspace

I am trying to prove that in a connected space - $X$ , for every two elements of $X$ - say $a,b$ I can find a subspace of $X$ ( say $X'$ ) , such that$ X'$ contains a,b and is also connected, and ...
1
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1answer
47 views

Connected Pontryagin dual

The dual group to a compact abelian group is discrete so in particular very much disconnected. I was trying to invent an example of a connected locally compact abelian group with connected dual which ...
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0answers
39 views

Proving an attractor (i.e set with self similarity) is connected

Let $K$ be an attractor for iterating function system of two similarity maps i.e $$K=f_1(K)\cup f_2(K)$$ A similarity map is defined to be $f_i:\mathbb{R}^d\to \mathbb{R}^d$ s.t $$\forall x,y\in ...
1
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1answer
33 views

Examples of non-compact connected spaces with the property…

I am looking for a non-compact connected space $X$ such that for any two disjoint closed $A,B\subseteq X$ there exists a proper closed connected $C\subseteq X$ such that $A\cup B\subseteq C$. I ...
0
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1answer
35 views

How to justify the statement that a graph is connected?

Is the graph connected? Justify. Because there is a path connecting all pairs of vertices, this graph is therefore connected? Is that right?
0
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2answers
61 views

Showing that $\mathbb{C}$\ {$x+iy|x,y\in \mathbb{Q}$} is connected

This I find it really hard to solve. I suppose the set {$x+iy|x,y\in \mathbb{Q}$} is neither closed or open. But I just cannot seem to find a way to go forward. Can someone help me out. Thanks
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0answers
80 views

Connected space such that (almost) no subspace is connected [duplicate]

Is there a connected space $(X,\tau)$ such that $X$ has more than $2$ points and the only proper connected subsets of $X$ are the singletons?
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1answer
36 views

Let X be a subspace of $\mathbb R^2$ consisting of points that at least one is rational. Prove that X is path-connected.

Let X be a subspace of $\mathbb R^2$ consisting of points such that at least one of coordinates x and y is rational. Prove that X is path-connected. A sketch is as follows. Is it right? Also How to ...
1
vote
1answer
27 views

Connectedness of sets

Let $E \subset S$. Suppose $E$ is not connected. Then in the induced topology of $E$ relative to $S$, $E$ and $\emptyset$ are not the only clopen sets. I can show using the above definition that if ...
6
votes
2answers
71 views

Path Connectedness and fixed points

We have the following given to us, Let $α, β \colon [0, 1] \to [0, 1]$ be (not necessarily continuous) functions such that $α(x) ≤ β(x)$, for all $x ∈ [0, 1]$. The set $K = \{\,(x, y); α(x) ≤ y ≤ ...
2
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1answer
32 views

Another example of a connected but non path connected set

I'm looking at the polar graph of the function $f:[\frac\pi2,\infty)\to \Bbb{R}^2$ defined by $\ r=f(\theta)=e^{\frac 1 \theta}$, the graph of this is the set of points of the form $(e^{\frac 1 ...
3
votes
3answers
78 views

k-Cells are Connected

I am studying real analysis from Baby Rudin, and while the book proves that real intervals are connected, it does not say anything regarding k-cells. I would expect them to also be connected, but do ...
8
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2answers
125 views

metric characterization for connectedness

Is there a metric characterization of connectedness? I'm looking for something like the following metric characterization of compactness: A metrizable topological space is compact if, and only if, ...
0
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1answer
11 views

Connected spaces of $M(n,\mathbb R)$

Consider $M(n,\mathbb R)$, the space of all $ n\times n$ matrices over $\mathbb R$.Which of the following are connected? a.$O(n)$ the set of all orthogonal matrices b.$GL(n,\mathbb R)$ set of all ...
0
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0answers
21 views

Least graph containing every connected graph with $m$ nodes as an induced subgraph

What is the smallest graph that contains every connected graph with $m$ nodes as an induced subgraph ? If the graph has $n$ nodes, there are $\binom{n}{m}$ (not necessariliy distinct) subgraphs with ...
0
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2answers
54 views

Is $\{(x,y)|y=\sin(\frac{1}{x})\}\cup(0,1)$ connected on $R^2$? [duplicate]

Source of definition: http://mathworld.wolfram.com/ConnectedSet.html Definition: A connected set is a set that cannot be partitioned into two nonempty subsets which are open in the relative topology ...
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0answers
35 views

A plane triangulation is 3-connected: Proof

I want to prove: "A plane triangulation $G$ with at least 4 vertices is 3-connected" I have found this proof. I don't like it but I took some ideas out of it: ...
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0answers
48 views

Mankiewicz theorem

I'm looking for a proof of Mankiewicz theorem, which states that: If $U, V$ are open, connected subsets of normed spaces $E, F$ respectively, then every bijective isometry $U \rightarrow V$ extends ...
1
vote
1answer
33 views

For planar triangulation, equivalence between 4-connectedness and non existence of separating triangle.

I want to prove the following equivalence: "A planar triangulation is 4-connected if and only if it has no separating triangle." My attempts so far: $\Rightarrow$: If there is a separating ...
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2answers
53 views

If $A$ is connected, is at least one of the sets $\mathrm{Int}A$ and $\mathrm{Bd}A$ connected?

If $A$ is a connected subset of $X$, does it follow at least one of the sets $\mathrm{Int}A$ and $\mathrm{Bd}A$ are connected? I have found counterexamples showing that they not both need to be ...
0
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1answer
22 views

Connected set is subset of the collection of its accumulation points.

A question from my analysis homework: Let $A \subset M$ be connected and contain more than one point. Show that every point in $A$ is an accumulation point of $A$. This question makes sense to me ...
0
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1answer
27 views

Relation between connected subset and clopen subset of a metric space?

I've read that for $A$ a connected subset of a metric space $M$ and $C$ a clopen (closed and open) subset of $M$, one could prove that either $A \subset C$ or $A \cap C=\varnothing$ and use it to ...
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0answers
44 views

$A$ is an interval so $A$ is connected?

I want to prove that if $A\subset \mathbb{R}$ is an interval then $A$ is connected. I found this proof, and I don't understand it essentially the ii) Suppose that $A$ is an interval but not ...
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0answers
12 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 ...
0
votes
1answer
28 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 $T$? ...
0
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1answer
27 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. ...
7
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0answers
97 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) $S$ ...