π-base: IDs 500-599

558

Has a cut point $\implies$ Cardinality $\geq 3$

Added:

Mar 12, 2026

Difficulty:

In order for $X \setminus \{p\}$ to be connected, it needs to have at least 2 points.

564

Locally finite $\implies$ Locally countable

Added:

Mar 12, 2026

Difficulty:

Finite implies countable.

567

(Hereditarily connected ∧ Locally finite) $\implies$ Countable

Added:

Mar 13, 2026

Difficulty:

$X$ is a countable union of finite open sets. $X$ hereditarily connected, so this basis is linearly ordered and it’s possible to enumerate them as $\{B_n\}$ where $n < m \implies B_n \subset B_m$ . Therefore, $B_n \nearrow X$ and $X$ is a union of finite sets.

571

Almost discrete $\implies$ ¬ Discrete

Added:

Mar 12, 2026

Difficulty:

It’s almost… so not quite.

584

Contractible $\implies$ ¬ Empty

Added:

Mar 12, 2026

Difficulty:

The homotopy cannot be empty.

IDs 500-599

Has a cut point ⟹ \implies⟹ Cardinality ≥3\geq 3≥3

Locally finite ⟹ \implies⟹ Locally countable

(Hereditarily connected ∧ Locally finite) ⟹ \implies⟹ Countable

Almost discrete ⟹ \implies⟹ ¬ Discrete

Contractible ⟹ \implies⟹ ¬ Empty

Has a cut point $\implies$ Cardinality $\geq 3$

Locally finite $\implies$ Locally countable

(Hereditarily connected ∧ Locally finite) $\implies$ Countable

Almost discrete $\implies$ ¬ Discrete

Contractible $\implies$ ¬ Empty