- #1
Demon117
- 165
- 1
1. Let M be a compact metric space. If r>0, show that there is a finite set S in M such that every point of M is within r units of some point in S.
2. Relevant theorems & Definitions:
-Every compact set is closed and bounded.
-A subset S of a metric space M is sequentially compact if every sequence in S has a subsequence that converges to a limit in S.
-If every open covering of S reduces to a finite subcovering then we say that S is covering compact.
3. Attempt at the solution
To begin with I don't completely understand coverings. So I am unsure whether I should use the sequential definition or the covering definition. Could someone please explain coverings?
I also wondered if S is a subset of a compact metric space, does it follow that S is compact?
INFORMAL PROOF: Since M is compact, it follows that every sequence in M has a convergent subsequence in M. Need to show that for any point in M is within some distance from a point in S. Let [tex]m_{k_{j}}[/tex] be such a subsequence. So for [tex]\epsilon>0[/tex] there exists N such that [tex]m_{k_{j}}[/tex] is within [tex]\epsilon[/tex] units of some point, call it m.
Not sure where to go from here.
2. Relevant theorems & Definitions:
-Every compact set is closed and bounded.
-A subset S of a metric space M is sequentially compact if every sequence in S has a subsequence that converges to a limit in S.
-If every open covering of S reduces to a finite subcovering then we say that S is covering compact.
3. Attempt at the solution
To begin with I don't completely understand coverings. So I am unsure whether I should use the sequential definition or the covering definition. Could someone please explain coverings?
I also wondered if S is a subset of a compact metric space, does it follow that S is compact?
INFORMAL PROOF: Since M is compact, it follows that every sequence in M has a convergent subsequence in M. Need to show that for any point in M is within some distance from a point in S. Let [tex]m_{k_{j}}[/tex] be such a subsequence. So for [tex]\epsilon>0[/tex] there exists N such that [tex]m_{k_{j}}[/tex] is within [tex]\epsilon[/tex] units of some point, call it m.
Not sure where to go from here.