Topology without Tears - 1.1 Definitions and Propositions

Definition:

Let \(X\) be a non-empty set. A set \( \tau\) is of subsets of \(X\) is said to be a topology on \( X \) if

1. \(X\) and the empty set, \( \emptyset \), belong to \( \tau \)
2. The union of any (finite (or) infinite) number of sets in \( \tau\) belongs to \( \tau\)
3. The intersection of any two sets in \( \tau\) belongs to \( \tau\)

The pair \( (X,\tau) \) is called a topological space.

Definition:

Let \(X\) be any non-empty set and let \(\tau\) be the collection of all subsets of \(X\). Then \(\tau\) is called the discrete topology on the set \(X\). The topological space \( (X,\tau) \) is called a discrete space.

Definition:

Let \(X\) be any non-empty set and let \(\tau\) be \( \{X, \emptyset\} \). Then \(\tau\) is called the indiscrete topology on the set \(X\). The topological space \( (X,\tau) \) is called a indiscrete space.

Proposition:

If \( (X,\tau) \) is a topological space such that, for every \( x \in X\), the singleton set \( \{x\}\) is in \( \tau\), then \( \tau\) is the discrete topology.

Proof:

Consider any \(A \subseteq X\). Note that \( A = \{x \in X : x \in A \} = \bigcup_{x \in A} \{x\} \). Since \( \tau\) is a topology and by the definition of a topology, any union must belong to the topology and hence \( A = \bigcup_{x \in A} \{x\} \in \tau\). Hence, we get that any subset of \(X\) belongs to \(\tau\). Hence, \( \tau\) is a discrete topology. \(\Box\)