3.1 Definition and Examples of Compact Spaces
In order to define a compact topological space, the notion of an open covering will be required. This is
defined as follows.
Definition 3.1.1. Let \(X\) be a topological space and let \(A\) be a subspace of \(X\). A collection \(\{A_{\lambda }:\lambda \in \Omega \}\) of subsets
of \(X\) is said to cover \(A\) if and only if every point in \(A\) belongs to atleast one of these subsets.
Definition 3.1.2. Let \(X\) be a topological space. A collection \(U=\{U_{\lambda }:\lambda \in \Omega \}\) of open subsets of \(X\) is said to be an
open covering of \(X\) if \(U\) covers \(X\). If \(U\) and \(V\) are open coverings of \(X\), then \(V\) is said to be a subcover of \(U\) if
every open set belonging to \(V\) also belongs to \(U\).
Definition 3.1.3. A topological space \(X\) is said to be compact if every open covering of \(X\) has a
finite sub-covering.
Proof. Consider the collection of open intervals \((\frac {1}{n},1)\) for all integers \(n\geq 2\). Each of these is open in (0,1),
and \((0,1)=\bigcup ^{\infty }_{n=2}(\frac {1}{n},1)\). But no finite sub-collection of this will suffice to cover (0,1), since the union of any finite
sub-collection \(\{(\frac {1}{n_1},1), (\frac {1}{n_2},1),..........,(\frac {1}{n_k})\}\) is just \((\frac {1}{N})\) where \(N=\max \{n_1,n_2,..........,n_k\}\)
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