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

1.1

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.

A1A2

1.2

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 $T_{d i s c}$ .

A1L2

1.3

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.

A1A2

1.4

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.

A1A2

1.5

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.

A1L1

2.1

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 ${\emptyset, X}$ .

step 1.2A1L1

2.2

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.

step 1.1step 1.2A2

2.3

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

step 1.3step 1.4A2

3.1

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

step 1.5step 2.1∎

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.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Discrete space (Wikipedia) (standard reference, not scraped)
Trivial topology (Wikipedia) (standard reference, not scraped)
J. Munkres, Topology, 2nd ed., §12 (standard reference, not scraped)