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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50 views

“Standard” proof that open disks in $\mathbb{R}^2$ are connected?

Homework for a complex analysis course asks me to prove as homework that open disks are connected. I do know a way to do this: open disks are convex, and an old exercise in Rudin's "Principles of ...
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0answers
22 views

Connectedness ( cardinality and connectedness) [duplicate]

$(X,d)$ metric space and $A\subset X$ and $A$ is connected. $$ \text{Card}(A) > 2 \implies \text{Card}(A) \geq \text{Card}(\mathbb{R}).$$ How do I prove it ?Waiting for your help?
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1answer
21 views

What is an example of a connected subset of $\mathbb{R}^2$ where the interior is not connected?

In $\mathbb{R}^2$ with the usual metric, could this be an open disk, e.g. $ \{(x,y) : x^2 + y^2 \le 2\}$? Thanks in advance!
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0answers
24 views

Coverings maps of a simply connected space

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 ...
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2answers
20 views

Condition that a local homeomorphism be a covering map.

Let be $f:Y\to X$ a local homeomorphism, with $Y$ a compact space and $X$ a Hausdorff connected space. How can I show that, for each $x\in X$, $p^{-1}(x)\subset Y$ is finite? So, is clear that $f$ is ...
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1answer
36 views

$A$ and $B$ are connected subsets in a metric space X. Prove at least one of $ A\cup B $ or $ A\cap B $ is connected.

$A$ and $B$ are connected subsets in a metric space X. Prove at least one of $ A\cup B $ or $ A\cap B $ is connected. I'm not sure where to start for this one. All I know about multiple connected ...
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2answers
50 views

Show that a set is connected and contractible?

I am having trouble with a practice qual exam question: Let $X = \{x,y\}$ with $\emptyset,X,\{x\}$ open. Show that X is connected and contractible? For the first part, I would assume not. That ...
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1answer
19 views

$X$ contains at least two points & at least one isolated point. Prove $X$ is not connected.

Can we take two sets $G_1 = (x_1)$, where $x_1$ is the isolated point, and $G_2 = B(x_2;\epsilon)-(x_2)$ where $x_2$ is a limit point and show that the set- connectedness conditions hold? Help would ...
4
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1answer
50 views

Complement of a point of a Compact Connected Hausdorff Space has no compact maximal connected subspace

This question is a slight modified version of Compact Connected Hausdorff Space has no compact component in the complement of a point Let $X$ be a Hausdorff Compact Connected Space. Prove that ...
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3answers
61 views

Can I find the connected components of a graph using matrix operations on the graph's adjacency matrix?

If I have an adjacency matrix for a graph, can I do a series of matrix operations on the adjacency matrix to find the connected components of the graph?
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1answer
56 views

Compact Connected Hausdorff Space has no compact component in the complement of a point

Let $X$ be a Hausdorff Compact Connected Space. Prove that $X\setminus\{x\}$ can't be expressed by the disjoint union of two connected sets with one them being compact.(lets assume the empty set is ...
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1answer
38 views

Check if the given set is Connected and Compact.

$S=\{\dfrac{x^{2}}{1+x^{2}}:x \in \mathbb R\}$ Since $S$ is not closed (the limit point $1$ does not belong to the set), so I concluded that $S$ is not compact. I am confused about verifying ...
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1answer
48 views

What are the components of the set of irrational numbers?

$X=\Bbb R$\ $ \Bbb Q $, standard topology What are the components of X? Is it for all $a, b \in \mathbb{R}$ and $c \in \mathbb{R} \setminus \mathbb{Q}$, for all $a <c<b$, (a, b) are the ...
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2answers
72 views

How to prove $ \mathit {X}$ is path connected?

${\mathbb{R}}^{2} $ Euclidean 2-space,Let $\mathit {X} \subset \mathbb{R}^{2} $.$$\mathit {X}=[-2,2]\times[-1,0]\cup[-2,-1]\times[0,1]\cup[1,2]\times[0,1]$$ $\qquad\qquad\qquad\qquad\qquad$ ...
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1answer
54 views

Why is $A = \{x \mid 1 < |x| < 2\}$ connected?

$A$ is $(-1, -2) \cup (1, 2)$, and these are two disjoint sets whose union makes up $A$, so it fits the definition of disconnected but the book says that $A$ is a domain (it is open and connected). ...
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2answers
127 views

Connectedness arguments in elementary mathematics?

To begin, let me explain a proof strategy (which I'll call the connectedness principle for want of a better, more canonical term): One way to prove that a mathematical object $O_1$ has some property ...
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0answers
15 views

connectedness image of a two variable function

Let $f:\Bbb{R}^2\rightarrow\Bbb{R}$ and two reals $a,b$ such that $$f (\mathbf{x,y})= \begin{cases} a(x^2+y^2),&(x,y)\in B_{d_2}(0;2)\\ \frac{b}{\sqrt{x^2+y^2}},&(x,y)\in \Bbb{R}^2- ...
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0answers
17 views

Is the quotient of connected linear algebraic groups connected?

Let $K$ be a field (of characteristic zero if it makes things nicer) and $G/K$ a connected linear algebraic group (i.e. smooth affine). Let $N \leq G$ be a connected normal linear $K$-subgroup of $G$. ...
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1answer
38 views

Show that the intersection of two connected sets is connected if the two sets are disjoint.

Show that the intersection of two connected sets is connected if the two sets are disjoint. Is the set $1\leq x^2+y^2+z^2 \leq 9$ connected and/or compact? I think its compact because it's closed ...
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2answers
26 views

Am I abusing set operations to justify an assumption in a simple proof involving the union of connected sets?

I'm working through some analysis textbooks on my own, so I don't want the full answer. I'm only looking for a hint on this problem. In Rosenlicht's Introduction to Analysis, he proves this ...
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1answer
52 views

$G^k$ is $k$-connected

For a connected graph $G$ on $n$ vertices and $1\le k \le n-1$ prove that the graph $G^k$ (where two vertices are connected if their distance is at most $k$) is $k$(-vertex)-connected. We need to ...
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0answers
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Can I change the proof this way?

In the proof I think he takes intersections of forward images. I am not that comfortable with this because I remember there was some problems with that. So I change it to inverse images. Is the proof ...
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1answer
60 views

Totally disconnected space in which some quasicomponents have interior?

Assume all spaces are metric. Question. Does there exist a space $X$ which is totally disconnected (the components of $X$ are singletons), yet some quasicomponent of $X$ has nonempty interior? I ...
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1answer
63 views

Property between totally disconnected and zero dimensional

Assume all spaces are Hausdorff. Definitions: $X$ is totally disconnected if the only nonempty connected subsets of $X$ are singletons. $X$ is zero dimensional if it has a base of clopen sets. ...
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4answers
67 views

If a continuous function satisfies $|f(z)^2-1|<1$ for every $z$, then either $|f(z)-1|<1$ of $|f(z)+1|<1$ for every $z$

Suppose a continuous function $f:D\rightarrow \mathbb{C}$ where $D$ is a plane domain, has the property $|f(z)^2-1|<1$ for every $z$ in $D$. Show that $|f(z)-1|<1$ of $|f(z)+1|<1$ for every ...
3
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2answers
57 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 ...
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2answers
69 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 ...
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1answer
21 views

Weakly Connected Graphs

How is the following graph a weakly connected graph?
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1answer
36 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 ...
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1answer
35 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 $D$ ...
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3answers
66 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.
3
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1answer
63 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 $X$, ...
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2answers
52 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
28 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), ...
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0answers
39 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 ...
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2answers
33 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 ...
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1answer
44 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 ...
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3answers
63 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
44 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 ...
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3answers
41 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 ...
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0answers
40 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 ...
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2answers
30 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 ...
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1answer
32 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
28 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 ...
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1answer
22 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 ...
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1answer
47 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 ...
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0answers
29 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] ...
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2answers
88 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 ...
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1answer
49 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
45 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 ...