Exercise 4.1 (basic test of openness). Suppose that $\mathcal{B}$ is a base of a topology on $X,$ and call the subsets of $X$ which are members of $\mathcal{B}$ 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 ${\mathbb{R}}^2,$ and let $\mathcal{Q}$ be the (countable) collection of all open squares in ${\mathbb{R}}^2$ where the coordinates of all four vertices are rational numbers. Prove that $\mathcal{Q}$ is a base for the Euclidean topology.

Deduce that the collection of all open sets in the Euclidean space ${\mathbb{R}}^2$ has cardinality $\mathrm{\aleph }$ (continuum), whereas the collection of all subsets of ${\mathbb{R}}^2$ has cardinality $2^{\mathrm{\aleph }}.$

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

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

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

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

Exercise 4.4 (subbasic test of continuity). Let $X,$ $Y$ be topological spaces, $f:X\to Y$ be a function, and $\mathcal{S}$ 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: $\mathrm{\forall }V\in \mathcal{S},$ $f^{-1}(V)$ is open in $X.$)

Exercise 4.5. (a) Let $X$ be a topological space and let $f:X\to \mathbb{R}$ be a function. Prove: $f$ is continuous iff for all $a,b\in \mathbb{R},$ 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 $f,g:X\to \mathbb{R}$ be continuous functions. Prove that the function $f+g:X\to \mathbb{R}$ is continuous. Hint: use (a).