Author |
Topic: Can one connected set pass through another? (Read 1080 times) |
|
ecoist
Senior Riddler
Gender: 
Posts: 405
|
 |
Can one connected set pass through another?
« on: Apr 4th, 2008, 9:24pm » |
 Quote  Modify
|
Let S be the square consisting of all points (x,y) in the plane such that 0<x,y<1. Can S be partitioned into two disjoint connected sets A and B such that A contains the vertical sides of S and B contains the horizontal sides of S?
|
|
 IP Logged |
|
|
|
Obob
Senior Riddler
Gender:
Posts: 489
|
|
Re: Can one connected set pass through another?
« Reply #1 on: Apr 5th, 2008, 9:35am » |
Quote Modify
|
By connected, do you mean path-connected?
|
|
IP Logged |
|
|
|
ecoist
Senior Riddler
Gender:
Posts: 405
|
|
Re: Can one connected set pass through another?
« Reply #2 on: Apr 5th, 2008, 10:15am » |
Quote Modify
|
No. The meaning here is: a set is connected if it is not the union of two disjoint, nonempty, open sets.
|
|
IP Logged |
|
|
|
Eigenray
wu::riddles Moderator
Uberpuzzler
Gender:
Posts: 1948
|
|
Re: Can one connected set pass through another?
« Reply #3 on: Apr 5th, 2008, 5:20pm » |
Quote Modify
|
I would guess yes, since we should be able to use the axiom of choice to eliminate each possible pair of separating sets.
|
|
IP Logged |
|
|
|
ecoist
Senior Riddler
Gender:
Posts: 405
|
|
Re: Can one connected set pass through another?
« Reply #4 on: Apr 5th, 2008, 9:15pm » |
Quote Modify
|
Sorry, I am topologically retarded. I think the right definition of connected set is: a set is connected if it is a set which cannot be partitioned into two nonempty subsets such that each subset has no points in common with the set closure of the other. The set of two distinct points in the plane is obviously not a connected set, yet it cannot be partitioned into two disjoint nonempty open sets. A point x is in the set closure of a set S if every neighborhood of x meets S.
|
|
IP Logged |
|
|
|
Hooie
Newbie
Gender:
Posts: 29
|
|
Re: Can one connected set pass through another?
« Reply #5 on: Apr 22nd, 2008, 6:02pm » |
Quote Modify
|
I believe you were right the first time. I think the two singletons would be open in the relative topology (a set U in a subspace A of X is open if it's of the form V intersect A, for some V open in X).
|
|
IP Logged |
|
|
|
ecoist
Senior Riddler
Gender:
Posts: 405
|
|
Re: Can one connected set pass through another?
« Reply #6 on: Apr 22nd, 2008, 8:49pm » |
Quote Modify
|
You are right, Hooie, but I wanted A and B to be connected in the plane, considered as a metric space. Hence, a set S in the plane is open if, for every point in S, some disk in the plane containing that point lies in S.
|
|
IP Logged |
|
|
|
Hooie
Newbie
Gender:
Posts: 29
|
|
Re: Can one connected set pass through another?
« Reply #7 on: Apr 23rd, 2008, 3:16am » |
Quote Modify
|
Yes, I know, but what it means to be connected depends on whether you're talking about the whole space IxI or the subset {p} U {q}.
A topological space X is connected if it can't be expressed as the disjoint union of two non-empty open subsets.
A subset Y of a topological space X is connected if it's connected in the relative topology it inherits from X.
|
|
IP Logged |
|
|
|
Hippo
Uberpuzzler
Gender:
Posts: 919
|
|
Re: Can one connected set pass through another?
« Reply #8 on: Sep 26th, 2008, 3:13pm » |
Quote Modify
|
What about set |y-sin (1/x)| < 0.01 (unioned with x=0 and with x=1)?
|
| « Last Edit: Sep 26th, 2008, 3:15pm by Hippo » |
IP Logged |
|
|
|
|
RIDDLES SITE
WRITE MATH!
Home 
Help 
Search 
Members 
Login 
Register
Reply
Notify of replies
Send Topic
Print