definition of exhaustion function on topological space
Topics
About: topological space
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
Starting Context
- The reader knows a definition of compact subset of topological space.
Target Context
- The reader will have a definition of exhaustion function on topological space.
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:
\( T\): \(\in \{\text{ the topological spaces }\}\)
\( \mathbb{R}\): \(= \text{ the Euclidean topological space }\)
\(*f\): \(: T \to \mathbb{R}\), \(\in \{\text{ the continuous maps }\}\)
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Conditions:
\(\forall r \in \mathbb{R} (f^{-1} ((- \infty, r]) \in \{\text{ the compact subsets of } T\})\)
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2: Natural Language Description
For any topological space, \(T\), and the Euclidean topological space, \(\mathbb{R}\), any continuous map, \(f: T \to \mathbb{R}\), such that for each \(r \in \mathbb{R}\), \(f^{-1} ((- \infty, r])\) is a compact subset of \(T\)
