definition of exhaustion of topological space by compact subsets
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.
- The reader knows a definition of interior of subset of topological space.
Target Context
- The reader will have a definition of exhaustion of topological space by compact subsets.
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:
\( T\): \(\in \{\text{ the topological spaces }\}\)
\(*s\): \(: \mathbb{N} \setminus \{0\} \to \{\text{ the compact subsets of } T\}\)
//
Conditions:
\(T = \cup_{j \in \mathbb{N} \setminus \{0\}} s (j)\)
\(\land\)
\(s (j) \subseteq {s (j + 1)}^\circ\), where \({s (j + 1)}^\circ\) denotes the topological interior of \(s (j + 1)\)
//
It is possible that \(s (j) = s (j + 1)\) for each \(j\) such that \(n \le j\) for an \(n \in \mathbb{N} \setminus \{0\}\), then, practically, \(s\) is finite. That means that \(s (j) = T\), so, \(T\) is compact. That is possible because when \(s (j + 1) = T\), \({s (j + 1)}^\circ = s (j + 1)\) because \(T\) is open.
2: Natural Language Description
For any topological space, \(T\), any sequence of subsets, \(s: \mathbb{N} \setminus \{0\} \to \{\text{ the compact subsets of } T\}\), such that \(T = \cup_{j \in \mathbb{N} \setminus \{0\}} s (j)\) and \(s (j) \subseteq {s (j + 1)}^\circ\), where \({s (j + 1)}^\circ\) denotes the topological interior of \(s (j + 1)\)
