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I am reading Stephen Willard: General Topology ... ... and am currently focused on 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:

In the above proof by Willard we read the following:

" ... ... First note that if \(\displaystyle A \subset B\), then by K-c, \(\displaystyle \overline{B} = \overline{A} \cup \overline{ (B -A) }\) so that \(\displaystyle \overline{A} \subset \overline{B}\) ... ... "

Can someone please demonstrate, formally and rigorously, how \(\displaystyle \overline{B} = \overline{A} \cup \overline{ (B -A) }\) implies that \(\displaystyle \overline{A} \subset \overline{B}\) ......

Help will be much appreciated ...

Peter

EDIT: it might be as simple as ... for two sets \(\displaystyle X\) and \(\displaystyle Y\) we have \(\displaystyle X \subset X \cup Y\) ... ... is that correct?

Peter

EDIT 2:

I am probably being pedantic ... but it might be more accurate to say that ...

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

... for sets \(\displaystyle X, Y\) and \(\displaystyle Z \) we have that ...

\(\displaystyle X = Y \cup Z \Longrightarrow Y \subset X\)... ...

Peter

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:

In the above proof by Willard we read the following:

" ... ... First note that if \(\displaystyle A \subset B\), then by K-c, \(\displaystyle \overline{B} = \overline{A} \cup \overline{ (B -A) }\) so that \(\displaystyle \overline{A} \subset \overline{B}\) ... ... "

Can someone please demonstrate, formally and rigorously, how \(\displaystyle \overline{B} = \overline{A} \cup \overline{ (B -A) }\) implies that \(\displaystyle \overline{A} \subset \overline{B}\) ......

Help will be much appreciated ...

Peter

EDIT: it might be as simple as ... for two sets \(\displaystyle X\) and \(\displaystyle Y\) we have \(\displaystyle X \subset X \cup Y\) ... ... is that correct?

Peter

EDIT 2:

I am probably being pedantic ... but it might be more accurate to say that ...

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

... for sets \(\displaystyle X, Y\) and \(\displaystyle Z \) we have that ...

\(\displaystyle X = Y \cup Z \Longrightarrow Y \subset X\)... ...

Peter

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