Topological Spaces and Continuity: Examples and Counterexamples

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✓ 13 results · all verified · 12 also independently AI-judged

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1 · Prerequisites

Real Analysis

Set Theory

Topology

2 · Logical flowchart

3 · Items

Example

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

Example

On an infinite set the cofinite topology has every infinite subset dense and no two nonempty open sets disjoint

Example

In the cocountable topology on ℝ the closed sets are the countable sets and ℝ, and a sequence converges iff it is eventually constant

Example

Sierpinski space and the particular-point topology, with their closures and their continuous maps

Example

The order topology on a totally ordered set, with the open rays as a subbasis, and its agreement with the usual topology of ℝ

Example

The Sorgenfrey line: ℝ with the half-open intervals [a,b) as a basis is strictly finer than the usual topology, is first countable, has a countable dense subset, and its sequences converge only from the right

Example

Closure and complement generate at most fourteen sets from any subset, and (0,1) ∪ (1,2) ∪ {3} ∪ ([4,5] ∩ ℚ) attains fourteen

Counterexample

In the cocountable topology on ℝ the sequential closure of [0,1] is [0,1] while its closure is all of ℝ

Counterexample

The identity from the cocountable topology on ℝ to the usual topology is sequentially continuous and not continuous

Counterexample

In the indiscrete topology every sequence converges to every point, and in the cofinite topology on an infinite set an injective sequence converges to every point

Counterexample

The identity from the discrete topology on ℝ to the usual topology is a continuous bijection that is not a homeomorphism

Counterexample

ℝ covered by its closed singletons: every restriction of the indicator of {0} is continuous and the map is not, so the closed pasting lemma needs finiteness

Counterexample

In ℝ the interiors of ℚ and of its complement are both empty while the interior of their union is everything

Counterexample

Topological Spaces and Continuity: Examples and Counterexamples

3 · Items

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

On an infinite set the cofinite topology has every infinite subset dense and no two nonempty open sets disjoint

In the cocountable topology on ℝ the closed sets are the countable sets and ℝ, and a sequence converges iff it is eventually constant

Sierpinski space and the particular-point topology, with their closures and their continuous maps

The order topology on a totally ordered set, with the open rays as a subbasis, and its agreement with the usual topology of ℝ

The Sorgenfrey line: ℝ with the half-open intervals [a,b) as a basis is strictly finer than the usual topology, is first countable, has a countable dense subset, and its sequences converge only from the right

Closure and complement generate at most fourteen sets from any subset, and (0,1) ∪ (1,2) ∪ {3} ∪ ([4,5] ∩ ℚ) attains fourteen

In the cocountable topology on ℝ the sequential closure of [0,1] is [0,1] while its closure is all of ℝ

The identity from the cocountable topology on ℝ to the usual topology is sequentially continuous and not continuous

In the indiscrete topology every sequence converges to every point, and in the cofinite topology on an infinite set an injective sequence converges to every point

The identity from the discrete topology on ℝ to the usual topology is a continuous bijection that is not a homeomorphism

ℝ covered by its closed singletons: every restriction of the indicator of {0} is continuous and the map is not, so the closed pasting lemma needs finiteness

In ℝ the interiors of ℚ and of its complement are both empty while the interior of their union is everything

The indiscrete topology on a two-point set is induced by no metric