5 Topology
A topology is a collection of subsets of a set that contains the empty set and the whole set and is closed under arbitrary unions and finite intersections that is used to define open sets and continuity of mappings.
definition [d]
A topology on a set \(X\) is a collection \(\tau\) of subsets of \(X\) such that
- \(\emptyset \in \tau\) and \(X \in \tau\)
- \(\tau\) is closed under arbitrary unions
- \(\tau\) is closed under finite intersections
where
- \(\emptyset\) is the empty set
- \(X\) is a set
- \(\tau\) is a collection of subsets of \(X\)
Note:
- The elements of \(\tau\) are called the open sets of the topology.
- A topology on \(X\) makes \(X\) into a topological space \((X,\tau)\).
5.1 Elementary Example
5.1.1 Simple
A topology on a three-point set lists which subsets are open. It must contain \(\emptyset\) and \(X\).
\[ X = \{ 1,\ 2,\ 3 \} \]
\[ \tau = \{ \emptyset,\ \{1\},\ \{1,2\},\ X \} \]
where
- \(X\) is the underlying set.
- \(\tau\) is the topology on \(X\).
- members of \(\tau\) are the open sets.
5.1.2 General
The same axioms on a four-point set: closed under unions and finite intersections.
\[ X = \{ a,\ b,\ c,\ d \} \]
\[ \tau = \{ \emptyset,\ \{a\},\ \{a,b\},\ \{a,b,c\},\ X \} \]
\[ \{a\} \cup \{a,b\} = \{a,b\} \in \tau,\quad \{a,b\} \cap \{a,b,c\} = \{a,b\} \in \tau \]
where
- \(\tau\) contains \(\emptyset\) and \(X\).
- \(\tau\) is closed under arbitrary unions and finite intersections.