Interior and Closure in a Topological Space .... .... remark by Willard

In summary, the conversation discusses the topic of closure and interior in topological spaces, specifically the duality between the two concepts as presented by Willard. The conversation talks about the definitions and results used by Willard and requests a formal and rigorous demonstration of the equation ##X-E^\circ=\overline{X-E}##. This is then followed by an explanation and example of how this equation can be derived using the definitions and results mentioned.
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I need help in order to prove a result stated by Willard linking the notions of interior and closure in a topological space ...
I am reading Stephen Willard: General Topology ... ... and am currently focused on Chapter 2: Topological Spaces and am currently focused on Section 3: Fundamental Concepts ... ...

I need help in order to fully understand a result or formula given by Willard concerning a link between closure and interior in a topological space ... ..The relevant text reads as follows:
Willard - Defn 3.9 plus remarks ... .png

In the above text by Willard we read the following:

" ... ... The strictly formal nature of this duality can be brought out in observing that

##X - E^{ \circ } = \overline{ X - E }## ... ... "Can someone please demonstrate (formally and rigorously) that ... given the definitions and results regarding closure and interior used by Willard ... ##X - E^{ \circ } = \overline{ X - E }## ... ...The definitions and results regarding closure used by Willard are as follows:
Willard - Defn 3.5, Lemma 3..6 and Theorem 3.7 .png


Help will be much appreciated ... ...

Peter
 
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Let's follow the definitions:
We have ##p\in X-E^\circ## if and only if every open set containing ##p## intersects ##X-E## nontrivially if and only if ##p## is in the closure of ##X-E.##

The first biconditional is just the negation of your definition of interior. The second is the following fact: Let ##X## be a topological space and let ##E\subset X##. Then ##p\in \overline{E}## if and only if every neighborhood of ##p## intersects ##E## nontrivially. Is this fact okay, or do you also want an argument here?

Edit: A little more direct argument using your definitions is:

##X-E^\circ=X-\bigcup_{G\subset E \ \text{open}} G=\bigcap_{G\subset E \text{ open}} (X-G)=\bigcap_{F\supset X-E \text{ closed}}F=\overline{X-E}.##
 
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Thanks so much Infrared ...

Working through your post just now ...

Very much appreciate your help ...

Peter
 
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1. What is the definition of interior in a topological space?

The interior of a subset A in a topological space X is the largest open set contained in A. It is denoted as Int(A).

2. How is the interior of a set related to the closure of the same set?

The interior and closure of a set are complementary concepts. The interior of a set A is the largest open set contained in A, while the closure of A is the smallest closed set containing A. In other words, the interior of A is contained in A, while A is contained in the closure of A.

3. Can the interior of a set be empty?

Yes, the interior of a set can be empty if the set itself is not open. For example, the set of rational numbers has an empty interior in the real number line.

4. What is the significance of the interior and closure in topological spaces?

The interior and closure are important concepts in topology as they help define and characterize different types of spaces. They are used to define open and closed sets, as well as to determine the convergence and continuity of functions on topological spaces.

5. How does the concept of interior and closure extend to metric spaces?

In metric spaces, the interior and closure are defined in a similar way as in topological spaces. However, the definition of open and closed sets is based on the notion of distance between points, rather than open and closed sets. The interior and closure of a set in a metric space can also be characterized using open and closed balls.

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