description/proof of that product of compact topological spaces is compact (Tychonoff theorem)
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of compact topological space.
- The reader knows a definition of product topological space.
- The reader admits the proposition that any topological space is compact if and only if each universal net with directed index set into the space is convergent.
- The reader admits the proposition that for any universal net, the composition of the net before any map into any another topological space is universal.
- The reader admits the proposition that any net to any product topological space converges to a point if and only if the projection to each constituent space after the net converges to the corresponding component of the point.
Target Context
- The reader will have a description and a proof of the proposition that the product of any possibly uncountable number of compact topological spaces is compact (the Tychonoff theorem).
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:
\(J\): \(\in \{\text{ the possibly uncountable index sets }\}\)
\(\{T_j \in \{\text{ the compact topological spaces }\} \vert j \in J\}\):
\(\times_{j \in J} T_j\): \(= \text{ the product topological space }\)
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Statements:
\(\times_{j \in J} T_j \in \{\text{ the compact topological spaces }\}\)
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2: Note
While the proposition that the product of any finite number of compact topological spaces is compact has been already proved, in fact, the product of any possibly uncountable number of compact topological spaces is compact, by this proposition.
3: Proof
Whole Strategy: Step 1: apply the proposition that any topological space is compact if and only if each universal net with directed index set into the space is convergent.
Step 1:
Let \(f: D \to \times_{j \in J} T_j\) be any universal net.
For each \(j \in J\), \(\pi^j \circ f: D \to T_j\), where \(\pi^j: \times_{j \in J} T_j \to T_j\) is the projection, is a universal net, by the proposition that for any universal net, the composition of the net before any map into any another topological space is universal.
\(\pi^j \circ f\) converges to an \(t_j \in T_j\), by the proposition that any topological space is compact if and only if each universal net with directed index set into the space is convergent, because \(T_j\) is compact.
Let \(t \in \times_{j \in J} T_j\) be such that for each \(j \in J\), the \(j\) component, \(t^j = t_j\).
\(f\) converges to \(t\), by the proposition that any net to any product topological space converges to a point if and only if the projection to each constituent space after the net converges to the corresponding component of the point.
So, \(\times_{j \in J} T_j\) is compact, by the proposition that any topological space is compact if and only if each universal net with directed index set into the space is convergent.
