The discrete and indiscrete topologies, their closures and interiors, and their continuous maps in each direction

Let $X$ be a set, let $\mathcal{T}_{\mathrm{disc}} = \mathcal{P}(X)$ be the discrete topology and $\mathcal{T}_{\mathrm{ind}} = \{\varnothing, X\}$ the indiscrete topology (The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies), and let $A \subseteq X$. Then:

1. In the discrete space every subset is clopen, and $\operatorname{int}(A) = A = \overline{A}, \qquad \partial A = \varnothing$ for every $A$ (Interior, closure, boundary, exterior, derived set and isolated point in a topological space). The singletons $\{\, \{x\} : x \in X \,\}$ form a basis (Basis and subbasis for a topology, and the topology generated by a family of sets).
2. In the indiscrete space $\operatorname{int}(A) = \begin{cases} X & A = X \\ \varnothing & A \ne X \end{cases}, \qquad \overline{A} = \begin{cases} \varnothing & A = \varnothing \\ X & A \ne \varnothing \end{cases},$ so $\partial A = X$ for every $A$ other than $\varnothing$ and $X$.
3. Maps out of a discrete space and into an indiscrete space are all continuous. For any topological space $Y$, every function $(X, \mathcal{T}_{\mathrm{disc}}) \to Y$ is continuous, and every function $Y \to (X, \mathcal{T}_{\mathrm{ind}})$ is continuous.
4. The other two directions are restrictive. A function $f : Y \to (X, \mathcal{T}_{\mathrm{disc}})$ is continuous exactly when $f^{-1}[\{x\}]$ is open in $Y$ for every $x \in X$; and a function $g : (X, \mathcal{T}_{\mathrm{ind}}) \to Y$ is continuous exactly when $g^{-1}[V] \in \{\varnothing, X\}$ for every open $V \subseteq Y$.

The two topologies are the extreme points of the comparison order (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison): every topology on $X$ is finer than $\mathcal{T}_{\mathrm{ind}}$ and coarser than $\mathcal{T}_{\mathrm{disc}}$.

• Both spaces are first countable, for opposite reasons. In the discrete space $\{\{x\}\}$ is a one-element neighbourhood base at $x$; in the indiscrete space $\{X\}$ is. So first countability is no obstruction to either, and it is the Hausdorff property that separates them: the discrete topology is metrizable by the metric taking the value $1$ on distinct points and $0$ on equal ones, whose ball of radius $1$ about $x$ is $\{x\}$, while the indiscrete topology on a set with two or more points is metrizable by none (The indiscrete topology on a two-point set is induced by no metric).

• Claim 3 explains why neither extreme is interesting on its own. A space in which every map is continuous carries no information about the maps; the content of a topology lies between the two extremes, and the comparison order of Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison is the scale on which that content is measured.

• In the indiscrete space every nonempty subset is dense by claim 2 and Dense, nowhere dense and codense subsets of a topological space, and the criterion by basic open sets, and in the discrete space only $X$ is. These are again the two extremes: density is a measure of how coarse the topology is, not of how large the set is.