Cheenta
How 9 Cheenta students ranked in top 100 in ISI and CMI Entrances?
Learn More

Compact Set, Proper Spaces and Annulus

Euclidean Spaces have a very nice property. In ( \mathbb{R}^n ) (equipped with standard Euclidean metric), every closed and bounded set is a compact set. The converse is also true. Every compact set is closed and bounded). This property is known as Heine Borel Theorem.

Recall that: A set V in a topological space X is compact iff every open cover of V has a finite subcover.

Sometimes, we want other spaces to have this property. Consider any metric space (X, d) with the property: every closed ball is compact. This type of space is known as proper metric space. 

We will prove a simple theorem related to proper spaces to illustrate their properties.

Theorem: Closed and Bounded Annulus in Proper Metric Spaces is Compact

Proof: Fix a point ( x_0 \in X ). Suppose ( B [ x_0 , r] = { x \in X | d(x_0, x) \leq r } ). In simpler terms, ( B[x_0, r] ) is a closed ball.

Let int (A) denote the interior points of the set A. (Recall that a point ( a\in A ) is called an interior point if we can find a neighborhood of a that is contained in A).

If a < b then (A =  B[x_0, b] - int (B[x_0, a] ) ) is a honest-to-goodness closed and bounded annulus. We will show that A is compact.

Suppose ( {U_{\alpha} }{\alpha \in \Lambda} ) is an arbitrary open cover of A. Then ( { {U{\alpha} }_{\alpha \in \Lambda} ,int (B[x_0, a] ) } ) is an open cover for ( B[x_0, b] ).

Since (X, d) is proper, by definition, ( B[x_0, b] ) is compact.

Hence a finite subclass of ( { {U_{\alpha} }_{\alpha \in \Lambda} ,int (B[x_0, a] ) } ) covers ( B[x_0, b] ). Since A is a subset of ( B[x_0, b] ), hence this finite subclass also covers A. Therefore we have found a finite subcover of A (for an arbitrary open cover of it).

This implies A is compact.

Also see:

Cheenta College Mathematics Program

Moral

This is a standard strategy. To show some set is compact, start with an arbitrary open cover of that set and find a finite subcover of it.

Knowledge Partner

Cheenta is a knowledge partner of Aditya Birla Education Academy
Cheenta

Cheenta Academy

Aditya Birla Education Academy

Aditya Birla Education Academy

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.
JOIN TRIAL
support@cheenta.com