Topology without Tears - 1.2 Definitions and Propositions

Definition:

Let \((X,\tau)\) be any topological space. Then the members of \(\tau\) are said to be open sets.

Proposition:

If \(X,\tau\) is any topological space, then

1. \(X\) and \(\emptyset\) are open sets.
2. The union of any (finite (or) infinite) number of open sets is an open set.
3. The intersection of any finite number of open sets is an open set.

Proof:

Clearly, \(1,2\) follow from the definitions of a topology. \(3\) follows from the definition of topology and problem \(4\) in the previous exercise. \(\Box\)

Definition:

Let \((X,\tau)\) be a topological space. A subset \(S \subset X\) is said to be a closed set in \((X,\tau)\) if its complement in \(X\), namely \(X \backslash S\), is open in \((X,\tau)\).

Proposition:

If \(X,\tau\) is any topological space, then

1. \(\emptyset\) and \(X\) are closed sets.
2. The intersection of any (finite (or) infinite) number of closed sets is a closed set.
3. The union of any finite number of closed sets is a closed set.

Proof:

• \(1\) follows immediately since \(X \backslash X = \emptyset \in \tau\) and \(X \backslash \emptyset = X \in \tau\).
• To prove \(2\), consider \(\bigcap_{\alpha \in \Gamma} A_{\alpha}\) where \(A_{\alpha}\) is a closed set i.e. \(X \backslash A_{\alpha} = X \bigcap A^c_{\alpha} \in \tau\). All we need to prove is \(X \bigcap \left(\bigcap_{\alpha \in \Gamma} A_{\alpha} \right)^c \in \tau\) i.e. we need to prove \(X \bigcap \left(\bigcup_{\alpha \in \Gamma}A_{\alpha}^c \right)\). Now since \(\tau\) is a topology and \(A_{\alpha}^c\) are open sets in \(\tau\), from property \(2\) of topology, we get \(\bigcup_{\alpha \in \Gamma} A_{\alpha}^c \in \tau\). Now from property \(3\) of topology, we get \(X \bigcap \left( \bigcup_{\alpha \in \Gamma} A_{\alpha}^c \right) \in \tau\). Hence, we get \(X \bigcap \left( \bigcap_{\alpha \in \Gamma} A_{\alpha}\right)^c \in \tau\) and hence, \(\bigcap_{\alpha \in \Gamma} A_{\alpha}\) is a closed set. Hence, the intersection of any (finite (or) infinite) number of closed sets is a closed set.
• To prove \(3\), consider closed sets \(\{A_k\}_{k=1}^{n}\). We need to prove that \(\bigcup_{k=1}^{n} A_k\) is also closed i.e. we need to prove that \(X \bigcap \left(\bigcup_{k=1}^{n} A_k \right)^c \in \tau\) i.e. we need to prove that \(X \bigcap \left( \bigcap_{k=1}^{n} A_k^c \right) \in \tau\) i.e. we need to prove that \(\bigcap_{k=1}^{n} \left( X \bigcap A_k^c \right) \in \tau\). Since \(A_k\) are closed by definition, we get \(X \bigcap A_{k}^c \in \tau\), \(\forall k \in \{1,2,\ldots,n\}\). Since \(\tau\) is a topology, using property \(3\) and problem \(4\) of the previous exercise, we get \(\bigcap_{k=1}^{n} \left( X \bigcap A_k^c \right) \in \tau\). Hence, the union of any finite number of closed sets is a closed set. \(\Box\)

Definition:

A subset \(S\) of a topological space \((X,\tau)\) is said to be clopen if it is both open and closed in \((X,\tau)\)