π-base: IDs 400-499

407

Metrizable $\implies$ Submetrizable

Added:

Mar 12, 2026

Difficulty:

A topology is a coarser topology of itself.

410

Has a coarser separable metrizable topology $\implies$ Submetrizable

Added:

Mar 12, 2026

Difficulty:

We’re just dropping the “separable”.

428

Cardinality $\geq 3$ $\implies$ Has multiple points

Added:

Mar 12, 2026

Difficulty:

Left as an exercise for the reader.

430

Cardinality $\geq 4$ $\implies$ Cardinality $\geq 3$

Added:

Mar 12, 2026

Difficulty:

If you have at least 4 apples, then you have at least 3 apples.

431

¬ Finite $\implies$ Cardinality $\geq 4$

Added:

Mar 12, 2026

Difficulty:

If you have an infinite amount of apples, then you have at least 4 apples.

446

Fixed point property $\implies$ ¬ Empty

Added:

Mar 12, 2026

Difficulty:

If it has a fixed point… it has… a point.

450

Indiscrete $\implies$ Second countable

Added:

Mar 12, 2026

Difficulty:

In general, a countable topology is second countable.

454

Countably infinite $\implies$ Countable

Added:

Mar 12, 2026

Difficulty:

Can’t argue with that.

455

Countably infinite $\implies$ ¬ Finite

Added:

Mar 12, 2026

Difficulty:

Can’t argue with that.

456

(Countable ∧ ¬ Finite) $\implies$ Countably infinite

Added:

Mar 12, 2026

Difficulty:

Can’t argue with that.

480

(Compact ∧ Connected ∧ $T_2$ ) $\implies$ Continuum

Added:

Mar 12, 2026

Difficulty:

Literally by definition.

481

Continuum $\implies$ $T_2$

Added:

Mar 12, 2026

Difficulty:

Literally by definition.

482

Continuum $\implies$ Compact

Added:

Mar 12, 2026

Difficulty:

Literally by definition.

483

Continuum $\implies$ Connected

Added:

Mar 12, 2026

Difficulty:

Literally by definition.

489

Ordinal space $\implies$ LOTS

Added:

Mar 12, 2026

Difficulty:

It’s an order topology.

493

(Countable ∧ Discrete) $\implies$ Ordinal space

Added:

Mar 12, 2026

Difficulty:

It has a bijection $f: X \to n$ if finite, or $f : X \to \omega$ if infinite. It’s a homeomorphism either way.

IDs 400-499

Metrizable ⟹ \implies⟹ Submetrizable

Has a coarser separable metrizable topology ⟹ \implies⟹ Submetrizable

Cardinality ≥3\geq 3≥3 ⟹ \implies⟹ Has multiple points

Cardinality ≥4\geq 4≥4 ⟹ \implies⟹ Cardinality ≥3\geq 3≥3

¬ Finite ⟹ \implies⟹ Cardinality ≥4\geq 4≥4

Fixed point property ⟹ \implies⟹ ¬ Empty

Indiscrete ⟹ \implies⟹ Second countable

Countably infinite ⟹ \implies⟹ Countable

Countably infinite ⟹ \implies⟹ ¬ Finite

(Countable ∧ ¬ Finite) ⟹ \implies⟹ Countably infinite

(Compact ∧ Connected ∧ T2T_2T2​) ⟹ \implies⟹ Continuum

Continuum ⟹ \implies⟹ T2T_2T2​

Continuum ⟹ \implies⟹ Compact

Continuum ⟹ \implies⟹ Connected

Ordinal space ⟹ \implies⟹ LOTS

(Countable ∧ Discrete) ⟹ \implies⟹ Ordinal space

Metrizable $\implies$ Submetrizable

Has a coarser separable metrizable topology $\implies$ Submetrizable

Cardinality $\geq 3$ $\implies$ Has multiple points

Cardinality $\geq 4$ $\implies$ Cardinality $\geq 3$

¬ Finite $\implies$ Cardinality $\geq 4$

Fixed point property $\implies$ ¬ Empty

Indiscrete $\implies$ Second countable

Countably infinite $\implies$ Countable

Countably infinite $\implies$ ¬ Finite

(Countable ∧ ¬ Finite) $\implies$ Countably infinite

(Compact ∧ Connected ∧ $T_2$ ) $\implies$ Continuum

Continuum $\implies$ $T_2$

Continuum $\implies$ Compact

Continuum $\implies$ Connected

Ordinal space $\implies$ LOTS

(Countable ∧ Discrete) $\implies$ Ordinal space