description/proof of that subnet of universal net with directed index set is universal
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of subnet of net with directed index set.
- The reader knows a definition of universal net with directed index set.
Target Context
- The reader will have a description and a proof of the proposition that any subnet of any universal net with directed index set is universal.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(D_2\): \(\in \{\text{ the directed sets }\}\)
\(T\): \(\in \{\text{ the topological spaces }\}\)
\(f_2\): \(: D_2 \to T\), \(\in \{\text{ the universal nets }\}\)
\(D_1\): \(\in \{\text{ the directed sets }\}\)
\(f_1\): \(: D_1 \to D_2\), \(\in \{\text{ the final maps }\}\)
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Statements:
\(f_2 \circ f_1 \in \{\text{ the universal nets }\}\)
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2: Proof
Whole Strategy: Step 1: see that \(f_2 \circ f_1\) satisfies the conditions to be universal.
Step 1:
Let \(S \subseteq T\) be any.
As \(f_2\) is universal, \(f_2\) is eventually in \(S\) or is eventually in \(T \setminus S\).
Let us suppose that \(f_2\) is eventually in \(S\).
There is a \(d_2 \in D_2\) such that for each \({d_2}' \in D_2\) such that \(d_2 \le {d_2}'\), \(f_2 ({d_2}') \in S\).
As \(f_1\) is final, there is a \(d_1 \in D_1\) such that for each \({d_1}' \in D_1\) such that \(d_1 \le {d_1}'\), \(d_2 \le f_1 ({d_1}')\), so, \(f_2 \circ f_1 ({d_1}') \in S\).
So, \(f_2 \circ f_1\) is eventually in \(S\).
Let us suppose that \(f_2\) is eventually in \(T \setminus S\).
\(f_2 \circ f_1\) is eventually in \(T \setminus S\), likewise: it is just replacing \(S\) with \(T \setminus S\).
\(f_2 \circ f_1\) is eventually in \(S\) or is eventually in \(T \setminus S\).
So, \(f_2 \circ f_1\) is universal.
