description/proof of that for metric space with induced topology, compact subset, and open cover of subset, there is positive real number (Lebesgue number) s.t. subset of subset with diameter smaller than number is contained in element of cover
Topics
About: metric space
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of topology induced by metric.
- The reader knows a definition of compact subset of topological space.
- The reader knows a definition of bounded subset of metric space.
Target Context
- The reader will have a description and a proof of the proposition that for any metric space with the induced topology, any compact subset, and any open cover of the subset, there is a positive real number (Lebesgue number) such that each subset of the subset with diameter smaller than the number is contained in an element of the cover.
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:
\(M\): \(\in \{\text{ the metric spaces }\}\), with the induced topology
\(S\): \(\in \{\text{ the compact subsets of } M\}\)
\(J\): \(\in \{\text{ the possibly uncountable index sets }\}\)
\(Q\): \(= \{U_j \in \{\text{ the open subsets of } M\} \vert j \in J\}\) such that \(S \subseteq \cup_{j \in J} U_j\)
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Statements:
\(\exists r \in \mathbb{R} \text{ such that } 0 \lt r (\forall S^` \subseteq S \text{ such that } Diam (S^`) \lt r (\exists j \in J (S^` \subseteq U_j)))\)
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2: Note
When \(M\) is compact, \(S = M\) is a compact subset, and for each \(Q\), there is an \(r\) such that for each \(S^` \subseteq M\) with diameter smaller than \(r\), there is a \(j \in J\) such that \(S^` \subseteq U_j\).
3: Proof
Whole Strategy: Step 1: take an open cover of \(S\), \(\{B_{s, \epsilon_s / 2} \vert s \in S\}\), such that \(B_{s, \epsilon_s} \subseteq U_j\), and take a finite subcover, \(\{B_{s_l, \epsilon_{s_l} / 2} \vert l \in L\}\); Step 2: take \(r := Min (\{\epsilon_{s_l} / 2 \vert l \in L\})\); Step 3: see that \(r\) satisfies the conditions for this proposition.
Step 1:
For each \(s \in S\), \(s \in U_j\) for a \(j \in J\), and there is a \(B_{s, \epsilon_s} \subseteq M\) such that \(B_{s, \epsilon_s} \subseteq U_j\), where \(\epsilon_s\) depends on \(s\).
\(\{B_{s, \epsilon_s / 2} \vert s \in S\}\) is an open cover of \(S\).
As \(S\) is compact, there is a finite subcover, \(\{B_{s_l, \epsilon_{s_l} / 2} \vert l \in L\}\).
Step 2:
Let us take \(r := Min (\{\epsilon_{s_l} / 2 \vert l \in L\})\).
Step 3:
Let us see that \(r\) satisfies the conditions for this proposition.
Let \(S^`\) be any.
When \(S^` = \emptyset\), for any \(j \in J\), \(S^` \subseteq U_j\).
Let us suppose otherwise, hereafter.
There is an \(s^` \in S^`\).
\(s^` \in B_{s_l, \epsilon_{s_l} / 2}\) for an \(l \in L\), but \(B_{s_l, \epsilon_{s_l} / 2} \subseteq B_{s_l, \epsilon_{s_l}} \subseteq U_j\) for a \(j \in J\).
For each \({s^`}' \in S^`\), \(dist (s_l, {s^`}') \le dist (s_l, s^`) + dist (s^`, {s^`}') \lt \epsilon_{s_l} / 2 + r \le \epsilon_{s_l} / 2 + \epsilon_{s_l} / 2 = \epsilon_{s_l}\), so, \({s^`}' \in B_{s_l, \epsilon_{s_l}} \subseteq U_j\).
So, \(S^` \subseteq U_j\).
