Linked Questions

3
votes
1answer
1k views

union-of-connected-subsets-is-connected-if-intersection-is-nonempty [duplicate]

Union of connected subsets is connected if intersection is nonempty I don't understand why A∩F and B∩F are relatively open where Brian Scott commented. Thanks
0
votes
1answer
213 views

Show that if the intersection is non-empty, then the union is connected of the following [duplicate]

Let $\{A_i\}_{i \in I}$ be a family of connected subsets of a metric space $X$ ($I$ is some set of indices). Show that if the intersection $\bigcap A_i \neq \emptyset$ , then $\bigcup A_i$ is ...
1
vote
2answers
99 views

How to finish? Connectedness [duplicate]

If $A$ is a connected set and $\{A_i : i \in I\}$, $I$ an arbitrary set (can be countable or not) of connected sets. How to show that if $A \cap A_i \neq \emptyset$ for all $i \in I$ then $A \cup (\...
0
votes
1answer
126 views

Complex Analysis: Show the union of 2 regions is connected [duplicate]

Let G1 and G2 be two regions. Suppose that G1 ∩ G2 ≠ 0. Show that G1 U G2 is connected. I know I have to first show that G1 U G2 is open and then show it is connected but I have no idea where to ...
0
votes
1answer
62 views

Proving the Daisy Lemma [duplicate]

Lemma: Suppose that $A,B \subseteq X$ are connected and $A \cap B \neq \emptyset$ , then $A \cup B$ is connected. How would I go about proving this? I think I understand the consequences of the lemma ...
0
votes
0answers
23 views

Prove a family of connected sets with one set intersecting all others is connected [duplicate]

The question: If $\{A_j : j \in S\}$ is a family of connected sets and if one set of the family, $A$ intersects all the others, prove that $X = \cup_{j \in S} A_j$ is connected. My attempt at the ...
3
votes
3answers
466 views

Union of connected sets which have pairwisely nonempty intersections

Let $E_\alpha$ be open for all $\alpha \in I$ such that $E_\alpha \cap E_{\beta} \neq \emptyset$, then $\displaystyle\bigcup_{\alpha \in I}E_\alpha$ is also connected. There are similar questions to ...
1
vote
1answer
565 views

How to show “If A and B connected, is $A\cup B$ connected”?

If A and B connected, is $A\cup B$ connected? or give a counterexample. I'd say no because when we take $A=[1,2]$, $B=[3,4]$, these closed intervals are connected. But when we take $U=]\frac 12,\frac ...
2
votes
3answers
70 views

Proving that $\{(x,y):x^2-2x+y^2=0\}\cup \{(x,0):x\in [2,3]\}$ is connected

Let $E=\{(x,y):x^2-2x+y^2=0\}\cup \{(x,0):x\in [2,3]\}$. It appears to me from the sketch that this set is connected. However, I have no idea how to prove this, since the set is not convex, and we ...
2
votes
2answers
88 views

Proving a set is connected in $\mathbb{R}^2$

Let $B = \{(x,y) \in \mathbb{R}^2 : x\in \mathbb{Q} \text{ or } y\in \mathbb{Q}\}$. Prove that $B$ is either connected or disconnected. I am not sure if this logic is correct but I believe its ...
0
votes
2answers
120 views

if union of connected subsets are connected in a metric space, what if the union is the metric space $X$ itself?

With reference to this link that i read: Union of connected subsets is connected if intersection is nonempty My question is: if union of connected subsets are connected in a metric space and their ...
2
votes
0answers
190 views

Arbitrary union of connected subsets Follands

$\textbf{Question:}$ If $\{E_{\alpha} \}_{\alpha \in A}$ is a collection of connected subsets of $X$ such that $\bigcap_{\alpha \in A} E_{\alpha} \neq \emptyset$, then $ \bigcup_{\alpha \in A} E_{\...
1
vote
2answers
68 views

$\mathbb{R}^{p}\backslash \{0 \}$ is connected for $p \geq 2$

I have to show that $\mathbb{R}^{p}\backslash \{0 \}$ is connected for $p \geq 2$. Is it possible to show this using the property debated on in this article: Union of connected subsets is connected if ...
0
votes
2answers
95 views

Elementary topology of $\mathbb C$: Union of 2 regions with nonempty intersection is a region

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Exer 1.31,32 Perhaps related but my topology is so far limited to the elementary topology in ...
1
vote
3answers
84 views

Given (X, ||•||) normed space, prove that only X itself and empty space are clopen.

I' d like to ask you for some help. I' ve to prove the problem stated in title, but without using the knowledge that normed space is connected.And I just got no idea how to do so... Thanks for any ...

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