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I am reading Stephen Willard: General Topology ... ... and am studying Chapter 2: Topological Spaces and am currently focused on Section 3: Fundamental Concepts ... ...
I need help in order to prove Theorem 3.11 Part 1a using the duality relations between closure and interior ... ..
The definition of interior and Theorem 3.11 read as follows:
Readers of this post necessarily need access to the "dual" theorem ... namely Theorem 3.7 ...
Theorem 3.7 (together with Willard's definition of closure and a relevant lemma) reads as follows:
So ... I need help in order to prove Theorem 3.11 1a assuming the dual result in Theorem 3.7 ( that is Ka or $A \subset \overline{A}$ ) using only the definitions of closure and interior and the dual relations: $X  A^{ \circ } = \overline{ X  A }$ and $X  \overline{ A} = ( X  A)^{ \circ }$ ...
My attempt so far is as follows:
To show $A^{ \circ } \subset A$ ...
Proof:
Assume $A \subset \overline{ A}$ ..
Now we have that ...
$A \subset \overline{ A}$
$\Longrightarrow X  \overline{ A} \subset X  A$
$\Longrightarrow (X  A)^{ \circ } \subset X  A$ ...
But how do I proceed from here ... ?
Help will be much appreciated ... ...
Peter
I need help in order to prove Theorem 3.11 Part 1a using the duality relations between closure and interior ... ..
The definition of interior and Theorem 3.11 read as follows:
Readers of this post necessarily need access to the "dual" theorem ... namely Theorem 3.7 ...
Theorem 3.7 (together with Willard's definition of closure and a relevant lemma) reads as follows:
So ... I need help in order to prove Theorem 3.11 1a assuming the dual result in Theorem 3.7 ( that is Ka or $A \subset \overline{A}$ ) using only the definitions of closure and interior and the dual relations: $X  A^{ \circ } = \overline{ X  A }$ and $X  \overline{ A} = ( X  A)^{ \circ }$ ...
My attempt so far is as follows:
To show $A^{ \circ } \subset A$ ...
Proof:
Assume $A \subset \overline{ A}$ ..
Now we have that ...
$A \subset \overline{ A}$
$\Longrightarrow X  \overline{ A} \subset X  A$
$\Longrightarrow (X  A)^{ \circ } \subset X  A$ ...
But how do I proceed from here ... ?
Help will be much appreciated ... ...
Peter
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