π-base: IDs 300-399

303

(Anticompact ∧ Compact) $\implies$ Finite

Added:

Mar 12, 2026

Difficulty:

A space is a subspace of itself.

306

(Scattered ∧ ¬ Empty) $\implies$ Has an isolated point

Added:

Mar 12, 2026

Difficulty:

If $X$ itself is not dense, then some point is isolated.

315

Empty $\implies$ Meager

Added:

Mar 12, 2026

Difficulty:

$X = \bigcup \emptyset$ where $\emptyset$ is countable and each $A \in \emptyset$ is nowhere dense.

326

Pseudometrizable $\implies$ Locally pseudometrizable

Added:

Mar 12, 2026

Difficulty:

Globally implies locally.

327

Locally metrizable $\implies$ Locally pseudometrizable

Added:

Mar 12, 2026

Difficulty:

Every metric is a pseudometric.

333

Topological $n$ -manifold $\implies$ $T_2$

Added:

Mar 12, 2026

Difficulty:

By definition.

335

$T_4$ $\implies$ Normal

Added:

Mar 12, 2026

Difficulty:

By definition.

336

$T_5$ $\implies$ Completely normal

Added:

Mar 12, 2026

Difficulty:

By definition.

337

Fully $T_4$ $\implies$ Fully normal

Added:

Mar 12, 2026

Difficulty:

By definition.

338

$T_6$ $\implies$ Perfectly normal

Added:

Mar 12, 2026

Difficulty:

By definition.

340

Topological $n$ -manifold $\implies$ Second countable

Added:

Mar 12, 2026

Difficulty:

By definition.

346

Has a group topology $\implies$ ¬ Empty

Added:

Mar 12, 2026

Difficulty:

Groups/monoids are nonempty by definition.

352

Has a countable $k$ -network $\implies$ Has a $\sigma$ -locally finite $k$ -network

Added:

Mar 14, 2026

Difficulty:

Any countable family is a countable union of finite sets. Finite sets are clearly locally finite. So any countable family is $\sigma$ -locally finite.

390

Cardinality $\leq\mathfrak c$ $\implies$ Cardinality $\leq 2^{\mathfrak c}$

Added:

Mar 12, 2026

Difficulty:

$\kappa < 2^\kappa$ is true for any cardinal.

IDs 300-399

(Anticompact ∧ Compact) ⟹ \implies⟹ Finite

(Scattered ∧ ¬ Empty) ⟹ \implies⟹ Has an isolated point

Empty ⟹ \implies⟹ Meager

Pseudometrizable ⟹ \implies⟹ Locally pseudometrizable

Locally metrizable ⟹ \implies⟹ Locally pseudometrizable

Topological nnn-manifold ⟹ \implies⟹ T2T_2T2​

T4T_4T4​ ⟹ \implies⟹ Normal

T5T_5T5​ ⟹ \implies⟹ Completely normal

Fully T4T_4T4​ ⟹ \implies⟹ Fully normal

T6T_6T6​ ⟹ \implies⟹ Perfectly normal

Topological nnn-manifold ⟹ \implies⟹ Second countable

Has a group topology ⟹ \implies⟹ ¬ Empty

Has a countable kkk-network ⟹ \implies⟹ Has a σ\sigmaσ-locally finite kkk-network

Cardinality ≤c\leq\mathfrak c≤c ⟹ \implies⟹ Cardinality ≤2c\leq 2^{\mathfrak c}≤2c

(Anticompact ∧ Compact) $\implies$ Finite

(Scattered ∧ ¬ Empty) $\implies$ Has an isolated point

Empty $\implies$ Meager

Pseudometrizable $\implies$ Locally pseudometrizable

Locally metrizable $\implies$ Locally pseudometrizable

Topological $n$ -manifold $\implies$ $T_2$

$T_4$ $\implies$ Normal

$T_5$ $\implies$ Completely normal

Fully $T_4$ $\implies$ Fully normal

$T_6$ $\implies$ Perfectly normal

Topological $n$ -manifold $\implies$ Second countable

Has a group topology $\implies$ ¬ Empty

Has a countable $k$ -network $\implies$ Has a $\sigma$ -locally finite $k$ -network

Cardinality $\leq\mathfrak c$ $\implies$ Cardinality $\leq 2^{\mathfrak c}$