COMPACT SETS IN METRIC SPACES : #compact #metric #spaces
Compact Sets In Metric Spaces :
Contents :
* Open Cover of a metric space
* Open Sub Cover of a metric space
* Compact Set
* Examples
Open Cover :
Let K be a non-empty subset of a metric space X. A collection {G⍺/⍺ꜪΔ} of open subsets of X is called an open cover of K if K⊆⋃G⍺.
For example : Consider a metric space (R,d) where R is the set of all real numbers and
let K=(0,1).
Put Gn=(1/n, 1-1/n) n=3,4,5,...
Then {Gn} is an open cover of K=(0,1).
Open subcover :
Suppose {G⍺/⍺ꜪΔ} is an open cover of a nonempty subset K of a metric space X. Then a finite collection {G⍺1,G⍺2,...,G⍺n} of open subsets of X is called an open subcover of the open cover {G⍺/⍺ꜪΔ} if K⊂G⍺1⋃G⍺2⋃...⋃G⍺n.
Compact set:
A non empty subset K of a metric space X is called compact if every open cover of K has a finite subcover.
Examples : 1. Consider a metric space (R,d) where R is the set of all real numbers Then K= [0,1]⊂ R is compact.
2. Earth is compact since because the earth is covered with infinite number of sand particles and water drops and then earth covered with 5 oceans and 7 continents. .
3. Human body is compact because human body is covered by infinite number of blood cells and also covered by finite number of parts i.e head, shoulders, two legs , two hands etc. 4. The sky is not compact since the sky is covered with infinite number of stars, galaxies, air particles, black holes, planets etc but the sky is not covered with any finite number of things.
properties of compact set :
1. compact subsets of metric spaces are closed.
2. Compact subsets of metric spaces are bounded.
2. Closed subsets of compact sets are compact.
3. Intersection of a closed and a compact set is compact.
4. Every K-cell is compact.
5. A nonempty subset E of the real line is compact iff it is closed and bounded
{ Heine-Borel Theorem}
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