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Math and science::Topology

Compactness

Compact sets can be thought of as a generalization of [what?]. See the reverse side for a justification.

There are multiple ways of formulating compactness, with some being more intuitive than others. 

There are 3 formulations that are particularly important.

1. Compactness, in terms of sequences in R.

A set KR is compact iff [what?].

What is the topological generalization of this definition?

The most basic example of such a closed set in R is [a what?].

Bolzano-Weierstrass's theorem connects the first and second formulations.

2. Compactness, in terms of [something], [something] sets.

A set KRn is compact iff it is [what and what?].

The next formulation is the most general of the three, graduating to the topic of topology. It uses the idea of a cover and a subcover.

Cover

Let X be a topological space. A cover of X is a family (Ui)iI of subsets of X such that [...]. It is finite iff the indexing set I is finite, and open iff Ui is open for each iI.

Given a cover (Ui)iI and JI, we say that [...] is a subcover of (Ui)iI if it is itself a cover of X.

3. Compactness, in terms of covers.

A topological space X is compact iff [what?].

What is the theorem that connects these three formulations of compactness?