Connectedness: Show Y U A & Y U B are Connected

  • Thread starter radou
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In summary: Lemma 23.1 says that a separation of Y \subset X is a disjoint pair of nonempty sets A, B whose union is Y, provided that neither of A, B contains a limit point of the other. This condition is necessary, but not sufficient, for A and B to be relatively open in Y. The examples I gave above satisfy the condition, but they're not disjoint.
  • #1
radou
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Homework Statement



This one is giving me serious trouble.

Let Y be a subset of X. Let both X and Y be connected. SHow that if A and B form a separation of X\Y, then Y U A and Y U B are connected.

The Attempt at a Solution



I know all the basic definitions and theorems from the chapter about connectedness preceding this exercise section, but any way I try it, I don't seem to get anywhere.

Any ideas?
 
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  • #2
When you say "[tex]A[/tex] and [tex]B[/tex] form a separation of [tex]X \setminus Y[/tex]", what does this mean, exactly? My guess would be what I usually call "[tex]\{A, B\}[/tex] disconnect [tex]X \setminus Y[/tex]", that is, [tex]A[/tex] and [tex]B[/tex] are disjoint (relatively) open subsets of [tex]X \setminus Y[/tex], whose union is [tex]X \setminus Y[/tex]. Is this correct?
 
  • #3
Well, partially. According to Munkres, there's a subtle difference:

If X is a topological space, a separation of X is a pair of non empty, disjoint and open subsets U and V of X whose union is X. (This is a definition.)

If Y is a subspace of X, a separation of Y is a pair of non empty disjoint sets U, V whose union if Y. (This is a Lemma)

Note that in the Lemma it is not required for the separation sets to be open.

So, I assume if we're talking about X\Y, which is a subspace of X, the sets in the separation are not required to be open.
 
  • #4
That won't work. You need some kind of relative openness hypothesis, otherwise the result is false. Take [tex]X = \mathbb{R}[/tex], [tex]Y = [0, 1][/tex], [tex]A = (-\infty, -1] \cup (1, 2)[/tex], [tex]B = (-1, 0) \cup [2, \infty)[/tex]. Then [tex]A[/tex] and [tex]B[/tex] form a separation (according to the conditions you gave) of [tex]X \setminus Y = (-\infty, 0) \cup (1, \infty)[/tex], but [tex]Y \cup A = (-\infty, -1] \cup [0, 2)[/tex] and [tex]Y \cup B = (-1, 1] \cup [2, \infty)[/tex] are both disconnected.

Also, with the definition of "separation" you cite for subspaces, you can give a separation for a connected subspace, such as [tex]X = \mathbb{R}, Y = [0, 1], U = [0, \textstyle\frac12), V = [\textstyle\frac12, 1][/tex]. That doesn't make sense.
 
  • #5
ystael said:
That won't work. You need some kind of relative openness hypothesis, otherwise the result is false. Take [tex]X = \mathbb{R}[/tex], [tex]Y = [0, 1][/tex], [tex]A = (-\infty, -1] \cup (1, 2)[/tex], [tex]B = (-1, 0) \cup [2, \infty)[/tex]. Then [tex]A[/tex] and [tex]B[/tex] form a separation (according to the conditions you gave) of [tex]X \setminus Y = (-\infty, 0) \cup (1, \infty)[/tex], but [tex]Y \cup A = (-\infty, -1] \cup [0, 2)[/tex] and [tex]Y \cup B = (-1, 1] \cup [2, \infty)[/tex] are both disconnected.

Also, with the definition of "separation" you cite for subspaces, you can give a separation for a connected subspace, such as [tex]X = \mathbb{R}, Y = [0, 1], U = [0, \textstyle\frac12), V = [\textstyle\frac12, 1][/tex]. That doesn't make sense.

Interesting, it's exactly what says in the book. So, in the "separation lemma" for subspaces, the sets should be open too?
 
  • #6
At least relatively open in the subspace, otherwise they are useless for establishing disconnectedness. I don't have a copy of the new edition of Munkres, so I can't guess what he might be thinking there.
 
  • #7
I found the solution to this exercise (Ex 23.12)

http://www.math.ku.dk/~moller/e02/3gt/opg/S23.pdf

Consider this problem solved, I'll go through it by myself, although I don't like the looks of it, for some reason.

I posted it if you happen to be interested to look at it regardless.
 
  • #8
ystael said:
At least relatively open in the subspace, otherwise they are useless for establishing disconnectedness. I don't have a copy of the new edition of Munkres, so I can't guess what he might be thinking there.

OK, I looked it up. The problem is that you missed an important hypothesis in the lemma 23.1, which changes things entirely.

Lemma 23.1 of Munkres says that a separation of [tex]Y \subset X[/tex] is a disjoint pair of nonempty sets [tex]A, B[/tex] whose union is [tex]Y[/tex], which satisfy the condition that neither of [tex]A, B[/tex] contains a limit point of the other. This latter condition is exactly what you need for [tex]A[/tex] and [tex]B[/tex] to be relatively open (in fact, clopen) in [tex]Y[/tex]. The examples I gave above do not satisfy the latter condition.
 
  • #9
ystael said:
OK, I looked it up. The problem is that you missed an important hypothesis in the lemma 23.1, which changes things entirely.

Lemma 23.1 of Munkres says that a separation of [tex]Y \subset X[/tex] is a disjoint pair of nonempty sets [tex]A, B[/tex] whose union is [tex]Y[/tex], which satisfy the condition that neither of [tex]A, B[/tex] contains a limit point of the other. This latter condition is exactly what you need for [tex]A[/tex] and [tex]B[/tex] to be relatively open (in fact, clopen) in [tex]Y[/tex]. The examples I gave above do not satisfy the latter condition.

You're right, for some reason I forgot to mention this.
 

Related to Connectedness: Show Y U A & Y U B are Connected

1. What is the definition of "connectedness" in scientific terms?

Connectedness in scientific terms refers to the relationship or link between two or more things. It can also indicate the degree to which these things are related or dependent on each other.

2. How do you determine if two objects, A and B, are connected?

To determine if two objects are connected, we can look at their physical proximity, the degree of interaction or influence between them, and any shared characteristics or properties. Additionally, mathematical and statistical methods can be used to quantify the level of connectedness between two objects.

3. Can two objects be connected even if they are not physically touching?

Yes, two objects can be connected even if they are not physically touching. For example, two research studies may be connected through shared findings or data, even though the researchers themselves may not have direct contact. Similarly, two individuals may have a strong emotional connection despite being physically apart.

4. What is the importance of studying connectedness?

Studying connectedness can help us better understand the relationships and interactions between different elements in a system. This can have practical applications in fields such as biology, ecology, and social sciences. Additionally, understanding connectedness can also aid in problem-solving and decision-making processes.

5. How can we visualize and represent connectedness in scientific studies?

There are various ways to visualize and represent connectedness in scientific studies, such as network diagrams, graphs, and mathematical models. These visual representations can help us see the connections and relationships between different elements in a system, making complex data easier to understand and analyze.

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