ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (z-ai/glm-5.2)audited 2026-07-27

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

Example

Let $X$ be a set, let $T_{d i s c} = P (X)$ be the discrete topology and $T_{i n d} = {\emptyset, X}$ the indiscrete topology (The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies), and let $A \subseteq X$ . Then:

In the discrete space every subset is clopen, and $int (A) = A = \overline{A}, \partial A = \emptyset$ 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).
In the indiscrete space $int (A) = {\begin{cases} X & A = X \\ \emptyset & A \neq X \end{cases}, \overline{A} = {\begin{cases} \emptyset & A = \emptyset \\ X & A \neq \emptyset \end{cases},$ so $\partial A = X$ for every $A$ other than $\emptyset$ and $X$ .
Maps out of a discrete space and into an indiscrete space are all continuous. For any topological space $Y$ , every function $(X, T_{d i s c}) \to Y$ is continuous, and every function $Y \to (X, T_{i n d})$ is continuous.
The other two directions are restrictive. A function $f : Y \to (X, T_{d i s c})$ is continuous exactly when $f^{- 1} [{x}]$ is open in $Y$ for every $x \in X$ ; and a function $g : (X, T_{i n d}) \to Y$ is continuous exactly when $g^{- 1} [V] \in {\emptyset, X}$ 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 $T_{i n d}$ and coarser than $T_{d i s c}$ .

Facts & Assumptions

Given: A set $X$ with the two topologies above, a subset $A \subseteq X$ , a topological space $Y$ , and functions $f : Y \to X$ and $g : X \to Y$ .

[A1]

$T_{d i s c} = P (X)$ and $T_{i n d} = {\emptyset, X}$ (The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies).

[A2]

$int (A)$ is the largest open subset of $A$ and $\overline{A}$ the smallest closed superset of $A$ ; $\partial A = \overline{A} ∖ int (A)$ ; a set is closed exactly when its complement is open (Interior, closure, boundary, exterior, derived set and isolated point in a topological space, Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison).

[L1]

A map is continuous exactly when preimages of open sets are open, and exactly when preimages of the members of any fixed basis are open, a basis being a subbasis for the topology it generates (For a map of spaces the following agree: continuity at every point, preimages of open sets open, preimages of closed sets closed, preimages of subbasic open sets open, and $f (\overline{A}) \subseteq \overline{f (A)}$ , clauses (b) and (d), Continuity of a map of topological spaces at a point and globally).

[L2]

A family $B$ of subsets of $X$ is a basis for a topology exactly when it covers $X$ and every point of an intersection of two members lies in a member inside that intersection; the topology is then the family of unions of subfamilies (A family is a basis for a unique topology iff it covers the set and every point of an intersection of two members lies in a member inside that intersection; finite intersections of any subbasis form a basis, Basis and subbasis for a topology, and the topology generated by a family of sets).

Verification

technique · direct

1.1A1A2

In the discrete topology every subset of $X$ is open by [A1], so every subset is also closed, its complement being open; hence every subset is clopen.

1.2A1L2

The singletons cover $X$ , and the intersection of two distinct singletons is empty while the intersection of a singleton with itself is that singleton; so the family of singletons satisfies the basis criterion, and the topology it generates consists of all unions of singletons, that is of all subsets of $X$ , which is $T_{d i s c}$ .

1.3A1A2

In the indiscrete topology the open subsets of $A$ are $\emptyset$ always and $X$ exactly when $A = X$ ; so $int (A) = X$ if $A = X$ and $int (A) = \emptyset$ otherwise.

1.4A1A2

In the indiscrete topology the closed sets are $\emptyset$ and $X$ , so the closed supersets of $A$ are $X$ always and $\emptyset$ exactly when $A = \emptyset$ ; hence $\overline{A} = \emptyset$ if $A = \emptyset$ and $\overline{A} = X$ otherwise.

1.5A1L1

For any function $h$ out of the discrete space and any open $V$ in the target, $h^{- 1} [V]$ is a subset of $X$ and hence open; for any function $h$ into the indiscrete space, the only open sets of the target are $\emptyset$ and $X$ , whose preimages are $\emptyset$ and the whole source, both open.

2.1step 1.2A1L1

For $f : Y \to (X, T_{d i s c})$ : the singletons form a basis by step 1.2, so by clause (d) of [L1] continuity of $f$ is exactly the openness of every $f^{- 1} [{x}]$ . For $g : (X, T_{i n d}) \to Y$ : by clause (b) continuity is exactly the condition that each $g^{- 1} [V]$ be open in the indiscrete topology, that is a member of ${\emptyset, X}$ .

2.2step 1.1step 1.2A2

By step 1.1 every $A \subseteq X$ is open and closed in the discrete topology, so $int (A) = A$ and $\overline{A} = A$ by [A2], whence $\partial A = \emptyset$ ; with step 1.2 this is claim 1.

2.3step 1.3step 1.4A2

Steps 1.3 and 1.4 are claim 2, and for $A \notin {\emptyset, X}$ they give $\partial A = X ∖ \emptyset = X$ .

3.1step 1.5step 2.1∎

Step 1.5 is claim 3 and step 2.1 is claim 4.

Remarks

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.

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

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources