1
vote
1answer
36 views

Connected subsets of metric (or T1) spaces

I have proved some statements about connected subsets of a metric space. They are really basic, but I want to make sure that they are true. Would someone please tell me whether these statements are ...
3
votes
1answer
42 views

Is for open connected $U$ the set $U_\varepsilon$ for small $\varepsilon$ connected?

Let for an open connected subset $U\subset \mathbb R^n$ and a number $\varepsilon >0$: $$ U_\varepsilon=\{x\in U: dist (x, \partial U)> \varepsilon \}. $$ Then $U_\varepsilon$ is open but in ...
0
votes
1answer
124 views

Hyperspace and connectedness

I'm looking for any theorems and proofs for connectedness for hyperspaces exp(X). I would like to take a look for especially this theorem: $$ X \textit{ is connected } \leftrightarrow exp(X) ...
0
votes
1answer
46 views

Finitely many connected components, prove interiors are also connected

Show that in a space with finitely many connected components $C_i, i = 1, ..., n$ their interiors $Int(C_i)$ are also connected. Is it true in general that the interior of a connected component is ...
0
votes
0answers
30 views

Constant Function over Connected, Compact Space

I am working on this problem and was wondering if I could get some feedback on my attempt at the proof. My gut tells me that I need a stronger argument as why my covering is actually a cover. I also ...
3
votes
3answers
77 views

Closed and Connected Subset of a Metric Space

My English may not be perfect since I'm not a native speaker, so please do point out the grammar mistakes if there are any. I've been reading Conway's "Functions of One Complex Variable", and ...
0
votes
1answer
72 views

Are there path-connected but not polygonal-connected sets?

This question came up in my mind. In the scope of normed spaces, does there exist a path-connected but not polygonal-connected set? I'd rather say no for open sets (my intuition is that ...
2
votes
0answers
92 views

Prove that $E$ is disconnected iff there exists two open disjoint sets $A$,$B$ in $X$

Let $(X,d)$ be a metric space. Prove that $E$ is disconnected iff there exists two open disjoint sets $A$,$B$ in $X$ such that $E\cap A\neq\emptyset, E\cap B\neq \emptyset$ and $E\subset A\cup B$. ...
1
vote
1answer
51 views

Connectedness and Continuity

Suppose $S^n\subset \mathbb{R}^n$ be the set of all points in $\mathbb{R}^n$ with unit Euclidean norm. That is $$S^n=\{x\in\mathbb{R}^n~:~\|x\|=1\}$$ If $f:S^{n-1}\rightarrow\mathbb{R},~n>2$, is ...
1
vote
1answer
41 views

Connected subspaces and homeomorphism

$S_{1} = \{ (x,y)| y=\sin (\frac{1}{x}), 0<x \le 1 \}$ $S_{2} = \{ (x,y)| y=\sin (\frac{1}{x}), -1\le x <0 \}$ Let $S= S_{1} \cup S_{2} \cup \{(0,0)\}$ 1) Is S connected as a subspace ...
3
votes
2answers
143 views

Any ball is connected?

Let $X$ be a compact , metric space. Assume that the closure of every each open ball it the closed ball with same center and radius. Prove that any ball in this space is connected.
0
votes
1answer
30 views

$(X,d)$ m.e., with $Y \subset X$: $Y$ is open, $Y$ is connected it's equivalent to another property.

Let $(X,d)$ be a metric space and let $Y \subset X$: $Y$ is open. Prove that $Y$ is connected if and only if there aren't $A,B \subset X$ non-empty such that $Y=A \cup B$ and $A \cap \overline ...
0
votes
1answer
78 views

Connected subsets of a metric space

I have to prove the following result: Suppose $X$ is a metric space, $Z$ is a metric subspace of $X$ and $S \subset Z$ Then $S$ is a connected subset of $X$ if, and only if,$S$ is a connected subset ...
0
votes
1answer
30 views

criteria for connectedness

I have to prove that if, $X$ is not the union of two disjoint non-empty closed subsets of itself, then, Either $X$ = $\phi$ or the only continuous functions from $X$ to the discrete space {${0,1}$} ...
2
votes
3answers
40 views

Question about criteria of connectedness

I have to prove the following : A) No proper non empty subset of $X$ is both open and closed in $X$ $implies$ that $X$ is not the union of two disjoint open subsets of itself. Attempt at the proof: ...
1
vote
1answer
46 views

Connected components of a given space

For every $n \in \mathbb N$, let $A_n=\{\frac{1}{n}\}\times[0,1]$, and let $X=\bigcup_{n \in \mathbb N} A_n \cup \{(0,0),(0,1)\}$. Prove that: i)$\{(0,0)\}$ and $\{(0,1)\}$ are connected components of ...
5
votes
3answers
148 views

Extending a connected open set

Assume $\emptyset\neq V\subseteq U\subseteq\mathbb{R}^n$ are open and connected sets so that $U\setminus\overline{V}$ is connected as well. Given any point $x\in U$, is there always a connected open ...
6
votes
2answers
362 views

Every open ball is connected

Let $(X,d)$ be a metric space such that for all $x \in X$ and all $r>0$, $\overline{B(x,r)} = \{y \in X \mid d(x,y)\leqslant r\}$ Show that every open ball of $X$ is connected. Note- I was trying ...
0
votes
1answer
268 views

Proving subset is not connected iff there exist open sets in X

Prove that E $\subseteq$ X is not connected if and only if there exist open sets $A, B \subseteq X$ such that $E \subseteq A ∪ B, A ∩ B$ = $\emptyset$ and $E ∩ A$ and $E ∩ B$ are both nonempty. $X$ ...
2
votes
3answers
80 views

If a connected set $A$ is contained in a set $B$ that is contained in $\mathrm{cl}(A)$, then $B$ must be connected

I don't follow the argument in line 3-4 that reads "Since $U$ is an open set containing $x$, there would be a point $y$ in $U \cap A$." If this line is indeed logical, could someone please elaborate ...
3
votes
1answer
107 views

Distance Between Subsets in Connected Spaces

Suppose $\langle X, d \rangle$ is a metric space. For any two sets $F,G \subseteq X$, by abuse of notation define $d(F,G) = \inf \{ d(f,g): f \in F, g \in G \}$. Let $\rho > 0$, $x \in X$, and ...
1
vote
1answer
118 views

The algebra of clopen sets vs. the algebra of connected components

Let $X$ be a topological space which, for my intents, may be assumed to be metrizable and compact if needed (let's say it's a closed subset of the unit cube or something like that). I know that: If ...
2
votes
1answer
97 views

Don't understand connectedness for Metric Spaces

My book says that a metric space $E$ is connected if the only subsets of $E$ that are both open and closed are $E$ and $\varnothing$. It goes on to say that if a space $E$ is not connected then it ...
0
votes
1answer
99 views

A result about connectedness and closed set.

Show that if $F$ is a closed and connected subset of a metric space $X$ then for every pair of points $a,b\in F$ and each $r>0$ there are points $z_0,z_1,\ldots,z_n$ in $F$ with $z_0=a$, $z_n=b$ ...
0
votes
1answer
54 views

Let $ f:(X, d) \mapsto (Y,d) $ be an mapping such that $ Graph (f) $ is connected. [duplicate]

Where $ X $ is connected. Does it imply $ f $ to be continuous?
4
votes
4answers
620 views

Product of connected spaces

You have two connected topological spaces $(A,B)$. Prove that $A\times B$ is also connected. I understand that I have to prove that there is a point in $B$ (call it $b$), that makes $A\times\{b\}$ ...
3
votes
2answers
155 views

Imposing the topology of open rays in $\Bbb R$

After having received Brian M. Scott's permission (see comments in the selected answer) I am integrating his suggestions with my own solutions to form a complete answer to the questions apperaing ...
1
vote
1answer
162 views

A question on quasi-components

I have been doing some reading on general topology, connectedness in particular. Here is a question on a topological concept called quasi-component. Here is a definition: ...
5
votes
1answer
570 views

Is $\mathbf{R}^\omega$ in the uniform topology connected?

Let $\mathbf{R}^\omega$ be the set of all (infinite) sequences of real numbers. Then is this space connected in the uniform topology? How to determine this? The uniform metric $p \colon ...
75
votes
2answers
3k views

Connected metric spaces with disjoint open balls

Let $X$ be the $S^1$ or a connected subset thereof, endowed with the standard metric. Then every open set $U\subseteq X$ is a disjoint union of open arcs, hence a disjoint union of open balls. Are ...
0
votes
1answer
127 views

Constructing sets from connected sets

I feel that this is probably really obvious, but I don't know how to get started. Is it true that every set in a metric space is the union of connected, pairwise-separated sets? And does this ...