Vipul: Created page with '{{topospace property implication| stronger = compact space| weaker = feebly compact space}} ==Statement== Any compact space is a feebly compact space. ==Definitions us…'

2009-12-24T01:16:02Z

Created page with '{{topospace property implication| stronger = compact space| weaker = feebly compact space}} ==Statement== Any compact space is a feebly compact space. ==Definitions us…'

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{{topospace property implication|
stronger = compact space|
weaker = feebly compact space}}

==Statement==

Any [[compact space]] is a [[feebly compact space]].

==Definitions used==

===Compact space===

{{further|[[Compact space]]}}

A topological space is termed compact if every [[open cover]] of the space has a finite subcover.

===Feebly compact space===

A topological space is termed feebly compact if every [[locally finite collection]] of nonempty [[open subset]]s is finite.

==Proof==

'''Given''': A compact space <math>X</math>, a locally finite collection of nonempty open subsets <math>V_i, i \in I</math> of <math>X</math>.

'''To prove''': <math>I</math> is finite.

'''Proof''':

# There exists a point-indexed open cover <math>U_x, x \in X</math> of <math>X</math> such that each <math>U_x</math> intersects finitely many <math>V_i</math>s: For each point <math>x \in X</math>, we can find an open subset <math>U_x</math> that works by the definition of locally finite. Putting these together, there exists a point-indexed open cover.
# There exists a finite subset <math>A</math> of <math>X</math> such that the <math>U_a, a \in A</math>, cover <math>X</math>: This follows from the previous step and the definition of compactness.
# The size of <math>I</math> is finite: Every <math>V_i,i \in I</math>, is a nonempty open subset of <math>X</math>, hence it intersects at least one of the <math>U_a</math>s. Thus, to count the <math>V_i</math>s, it suffices to count the <math>V_i</math>s that intersect at least one <math>U_a</math>. For each <math>U_a</math>, the number of <math>V_i</math>s intersecting it is finite, and there are finitely many <math>a</math>s in <math>A</math>, so the total number is finite.

Compact implies feebly compact - Revision history

Vipul: Created page with '{{topospace property implication| stronger = compact space| weaker = feebly compact space}} ==Statement== Any compact space is a feebly compact space. ==Definitions us…'