ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31

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In the cocountable topology on $R$ , a closure point outside $[0, 1]$ is reached by a net in $[0, 1]$ but by no sequence in $[0, 1]$

Example

Give $R$ the cocountable topology, let $A = [0, 1]$ , and let $p = 2$ . Then $p \in \overline{A}$ , hence a net in $A$ converges to $p$ , but no sequence in $A$ converges to $p$ .

Facts & Assumptions

Given: The cocountable topology on $R$ , $A = [0, 1]$ , and $p = 2$ .

[L1]

Nonempty cocountable opens have at most countable complements (The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies, Finite, countably infinite, countable, uncountable).

[L2]

$[0, 1]$ is uncountable (Every nondegenerate interval of $R$ is uncountable).

[L3]

A sequence converges only if it is eventually in every neighbourhood of its proposed limit (Convergence and cluster points of a sequence in a topological space, sequential continuity, and the sequential closure).

[L4]

A point lies in the closure of a subset exactly when some net in that subset converges to it (A point lies in the closure of a set if and only if a net in the set converges to it).

Verification

technique · constructive

1.1

Every neighbourhood $N$ of $2$ has at most countable complement, so it meets the uncountable set $A$ . Hence $2 \in \overline{A}$ , and [L4] supplies a net in $A$ converging to $2$ .

L1L2L4construct

1.2

Let $(a_{n})$ be a sequence in $A$ . Its range is at most countable and omits $2$ , so $R ∖ {a_{n} : n \in N}$ is a neighbourhood of $2$ containing none of its terms. Thus $(a_{n})$ does not converge to $2$ .

L1L3

2.1

The net from step 1.1 detects the closure point, whereas no sequence in $A$ does.

step 1.1step 1.2discharge-construct∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

38 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Cocountable topology (Wikipedia) (standard reference, not scraped)
Net (mathematics) (Wikipedia) (standard reference, not scraped)

In the cocountable topology on RR, a closure point outside [0,1][0,1] is reached by a net in [0,1][0,1] but by no sequence in [0,1][0,1]