Exercise 4.1 (basic test of openness). Suppose that \(\mathscr B\) is a base of a topology on \(X,\) and call the subsets of \(X\) which are members of \(\mathscr B\) basic open sets.

Let \(A\) be a subset of \(X.\) Prove that the following are equivalent:

• 1. \(A\) is open in \(X.\)

• 2. \(A\) is a union of a collection of basic open sets.

• 3. For each point \(x\in A,\) there exists a basic open set \(U\) such that \(x\in U\) and \(U\subseteq A.\)

Exercise 4.2 (the Euclidean topology has a countable base). Consider the Euclidean space \(\RR ^2,\) and let \(\mathscr Q\) be the (countable) collection of all open squares in \(\RR ^2\) where the coordinates of all four vertices are rational numbers. Prove that \(\mathscr Q\) is a base for the Euclidean topology.

Deduce that the collection of all open sets in the Euclidean space \(\RR ^2\) has cardinality \(\aleph \) (continuum), whereas the collection of all subsets of \(\RR ^2\) has cardinality \(2^\aleph .\)

Reminder about cardinal numbers:
• \(\aleph _0\) (aleph-zero) denotes the countably infinite cardinality, e.g., the cardinality of \(\NN ;\)
• \(\aleph \) (aleph) denotes the cardinality of continuum, e.g., the cardinality of \(\RR ,\)
• one has \(|\RR | = \aleph = 2^{\aleph _0} = |P(\NN )| > \aleph _0.\)

Exercise 4.3 (subbase). Let \((Y,\mathscr T)\) be a topological space. A subbase of \(\mathscr T\) is a collection \(\mathscr S\) of open sets such that finite intersections of sets from \(\mathscr S\) form a base of \(\mathscr T.\)

It is worth noting that, given any set \(Y\) (without topology) and any collection \(\mathscr S\) of subsets of \(Y,\) we can construct a topology \(\mathscr T_{\mathscr S}\) on \(X\) by using \(\mathscr S\) as a subbase. That is, \(\mathscr T_{\mathscr S}\) consists of arbitrary unions of finite intersections of members of \(\mathscr S.\) It is not difficult to show that this collection \(\mathscr T_{\mathscr S}\) is a topology.

Prove that the collection of all open rays in the real line, i.e., sets of the form \((-\infty ,a)\) and \((b,+\infty ),\) is a subbase of the Euclidean topology.

Exercise 4.4 (subbasic test of continuity). Let \(X,\) \(Y\) be topological spaces, \(f\colon X \to Y\) be a function, and \(\mathscr S\) be a subbase of topology on \(Y.\) Prove that the following are equivalent:

• 1. \(f\) is continuous.

• 2. The preimage of every subbasic set in \(Y\) is open in \(X\) (meaning: \(\forall V\in \mathscr S,\) \(f^{-1}(V)\) is open in \(X.\))

Exercise 4.5. (a) Let \(X\) be a topological space and let \(f\colon X\to \RR \) be a function. Prove: \(f\) is continuous iff for all \(a,b\in \RR ,\) the sets \(X_{f<a} = \{x\in X: f(x)<a\}\) and \(X_{f>b} ={\{x\in X}: f(x)>b\}\) are open in \(X.\)

(b) Let \(X\) be a topological space and let \(f,g\colon X\to \RR \) be continuous functions. Prove that the function \(f+g\colon X \to \RR \) is continuous. Hint: use (a).