π-base: IDs 100-199

106

(Lindelöf ∧ Countably compact) $\implies$ Compact

Added:

Mar 12, 2026

Difficulty:

Shrink the cover twice.

118

$T_2$ $\implies$ $T_1$

Added:

Mar 12, 2026

Difficulty:

Clear from their definitions.

119

$T_1$ $\implies$ $T_0$

Added:

Mar 12, 2026

Difficulty:

Clear from their definitions.

121

Compact $\implies$ $\sigma$ -compact

Added:

Mar 12, 2026

Difficulty:

A single set is trivially a countable union.

122

$\sigma$ -compact $\implies$ Lindelöf

Added:

Mar 12, 2026

Difficulty:

If each compact has a finite subcover, a countable union of them will have a countable subcover.

138

Cardinality $=\mathfrak c$ $\implies$ ¬ Cardinality $\lt\mathfrak c$

Added:

Mar 12, 2026

Difficulty:

Left as an exercise for the reader.

139

Cardinality $=\mathfrak c$ $\implies$ Cardinality $\leq\mathfrak c$

Added:

Mar 12, 2026

Difficulty:

Left as an exercise for the reader.

187

Finite $\implies$ Countable

Added:

Mar 12, 2026

Difficulty:

That’s right, “countable” does not mean infinitely countable.

189

Finite $\implies$ Second countable

Added:

Mar 12, 2026

Difficulty:

Any topology is a subset of $\mathcal{P}(X)$ , so there are finitely many open sets.

190

Cardinality $=\aleph_1$ $\implies$ Cardinality $\leq\mathfrak c$

Added:

Mar 12, 2026

Difficulty:

By definition, $\aleph_1$ is the smallest cardinality greater than $\aleph_0$ . Assuming the continuum hypothesis, $\aleph_1 = \mathfrak c$ .

191

Cardinality $=\aleph_1$ $\implies$ ¬ Countable

Added:

Mar 12, 2026

Difficulty:

By definition, $\aleph_1$ is the smallest uncountable cardinal.

198

Finite $\implies$ Noetherian

Added:

Mar 12, 2026

Difficulty:

Every subspace is finite.

IDs 100-199

(Lindelöf ∧ Countably compact) ⟹ \implies⟹ Compact

T2T_2T2​ ⟹ \implies⟹ T1T_1T1​

T1T_1T1​ ⟹ \implies⟹ T0T_0T0​

Compact ⟹ \implies⟹ σ\sigmaσ-compact

σ\sigmaσ-compact ⟹ \implies⟹ Lindelöf

Cardinality =c=\mathfrak c=c ⟹ \implies⟹ ¬ Cardinality <c\lt\mathfrak c<c

Cardinality =c=\mathfrak c=c ⟹ \implies⟹ Cardinality ≤c\leq\mathfrak c≤c

Finite ⟹ \implies⟹ Countable

Finite ⟹ \implies⟹ Second countable

Cardinality =ℵ1=\aleph_1=ℵ1​ ⟹ \implies⟹ Cardinality ≤c\leq\mathfrak c≤c

Cardinality =ℵ1=\aleph_1=ℵ1​ ⟹ \implies⟹ ¬ Countable

Finite ⟹ \implies⟹ Noetherian

(Lindelöf ∧ Countably compact) $\implies$ Compact

$T_2$ $\implies$ $T_1$

$T_1$ $\implies$ $T_0$

Compact $\implies$ $\sigma$ -compact

$\sigma$ -compact $\implies$ Lindelöf

Cardinality $=\mathfrak c$ $\implies$ ¬ Cardinality $\lt\mathfrak c$

Cardinality $=\mathfrak c$ $\implies$ Cardinality $\leq\mathfrak c$

Finite $\implies$ Countable

Finite $\implies$ Second countable

Cardinality $=\aleph_1$ $\implies$ Cardinality $\leq\mathfrak c$

Cardinality $=\aleph_1$ $\implies$ ¬ Countable

Finite $\implies$ Noetherian