description/proof of that projection of product topological space is not necessarily closed
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of product topological space.
- The reader knows a definition of closed map.
Target Context
- The reader will have a description and a proof of the proposition that a projection of a product topological space is not necessarily closed.
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:
\(J\): \(\in \{\text{ the possibly uncountable index sets }\}\)
\(\{T_j \in \{\text{ the topological spaces }\} \vert j \in J\}\):
\(\times_{j \in J} T_j\): \(= \text{ the product topological space }\)
\(l\): \(\in J\)
\(\pi^l\): \(: \times_{j \in J} T_j \to T_l, f \mapsto f (l)\), \(= \text{ the projection }\)
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Statements:
not necessarily "\(\pi^l \in \{\text{ the closed maps }\}\)"
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2: Note
It is not that \(\pi^l\) is inevitably non-closed.
For example, when \(T_1\) and \(T_2\) are discrete, \(\pi^l\) is closed, because each subset of \(T_l\) is closed.
Compare with the proposition that any projection of any product topological space is open.
3: Proof
Whole Strategy: Step 1: see an example that for a closed \(C \subseteq \times_{j \in J} T_j\), \(\pi^l (C)\) is not closed.
Step 1:
Let \(T_1 = \mathbb{R}\) and \(T_2 = \mathbb{R}\) be the Euclidean topological spaces.
The graph of \(tan: (- \pi / 2, \pi / 2) \subseteq T_1 \to T_2\), \(C \subseteq \mathbb{R}^2\), is closed.
But \(\pi^1 (C) = (- \pi / 2, \pi / 2) \subseteq \mathbb{R}\), which is not closed.
So, \(\pi^1\) is not closed.
