Closure in a Topological Space .... Willard, Theorem 3.7 .... ....

In summary, the conversation is discussing the proof of Theorem 3.7 in Stephen Willard's book on General Topology. The main focus is on the implication ##\overline{B} = \overline{A} \cup \overline{(B-A)} \Longrightarrow \overline{A} \subset \overline{B}##. The conversation ends with a final confirmation that this implication is true.
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TL;DR Summary
I need help in order to understand Willard Theorem 3.7 concerning topological closure ...
I am reading Stephen Willard: General Topology ... ... and am currently reading Chapter 2: Topological Spaces and am currently focused on Section 1: Fundamental Concepts ... ...

I need help in order to fully understand an aspect of the proof of Theorem 3.7 ... ..Theorem 3.7 and its proof read as follows:
Willard - 1 - Theorem 3.7 ... PART 1 ... .png

Willard - 2 - Theorem 3.7 ... PART 2 ... .png

In the above proof by Willard we read the following:

" ... ... First note that if ##A \subset B##, then by K-c, ##\overline{B} = \overline{A} \cup \overline{ (B -A) }## so that ##\overline{A} \subset \overline{B}## ... ... "Can someone please demonstrate, formally and rigorously, how ##\overline{B} = \overline{A} \cup \overline{ (B -A) }## implies that ##\overline{A} \subset \overline{B}## ...
Help will be much appreciated ...

Peter
 
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  • #2
Wow ... that was quick ...

Yes that's right ... hmm is it as simple as that ...

Thanks ...

Peter
 
  • #3
Hi. Something went wrong with wrong with the formatting. I think it follows from ##X\subseteq X\cup Y## for sets ##X,Y##.

(I deleted previous post, but it was correct).
 
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Thanks ...

I think I have corrected the formatting ...

I think you're correct ...

Thanks again ...

Peter
 
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Math_QED said:
Hi. Something went wrong with wrong with the formatting. I think it follows from ##X\subseteq X\cup Y## for sets ##X,Y##.

(I deleted previous post, but it was correct).
I am probably being pedantic ... but it might be more accurate to say that ...

##\overline{B} = \overline{A} \cup \overline{ (B -A) }## implies that ##\overline{A} \subset \overline{B}## ... ... is true because ...

... for sets ##X, Y## and ##Z## we have that ...

##X = Y \cup Z \Longrightarrow Y \subset X##... ...

Peter
 
  • #6
I would just write it like this:

##\overline{A}\subseteq \overline{A} \cup \overline{B - A} = \overline{B}##
 
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1. What is closure in a topological space?

Closure in a topological space refers to the set of all points that are either in the space itself or are limit points of the space. In other words, it is the smallest closed set that contains all the points in the space.

2. How is closure related to continuity?

The closure of a set is closely related to the concept of continuity in topology. A function is continuous if and only if the preimage of any closed set is also closed. This means that the closure of the image of a set under a continuous function is contained in the image of the closure of the set.

3. What is the significance of Theorem 3.7 in Willard's book?

Theorem 3.7 in Willard's book states that the closure of a set is equal to the intersection of all closed sets that contain the set. This theorem is important because it provides a way to determine the closure of a set without having to explicitly list all the limit points.

4. Can the closure of a set be empty?

Yes, it is possible for the closure of a set to be empty. This can happen if the set itself is empty or if all the points in the set are isolated points (i.e. they do not have any limit points).

5. How is closure different from boundary?

The closure of a set includes all the points in the set as well as its limit points, while the boundary of a set only includes its limit points. In other words, the closure is the union of the set and its boundary, while the boundary is the difference between the closure and the set itself.

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