π-base: IDs 1-99

1

Compact $\implies$ Countably compact

Added:

Mar 12, 2026

Difficulty:

Evident.

2

Countably compact $\implies$ Weakly countably compact

Added:

Mar 11, 2026

Difficulty:

Let $X$ be infinite and $A \subseteq X$ countably infinite (uses axiom of countable choice). If $X$ has no limit points, $A$ is closed and for each $x \in A$ , choose a nbd $V_x$ with $V_x \cap X = \emptyset$ . Then $\{V_n\} \cup \{A^C\}$ is a cover. A finite subcover would imply $A$ is finite.

4

Countably compact $\implies$ Pseudocompact

Added:

Mar 11, 2026

Difficulty:

For $f : X \to \R$ continuous, $\{f^{-1}(-n, n)\}$ is a countable cover.

6

Compact $\implies$ Locally relatively compact

Added:

Mar 12, 2026

Difficulty:

Any closed set is compact, so any closure of a nbd is compact.

7

Locally relatively compact $\implies$ Weakly locally compact

Added:

Mar 12, 2026

Difficulty:

Take one nbd from the local basis. Its closure is compact.

8

Exhaustible by compacts $\implies$ Weakly locally compact

Added:

Mar 12, 2026

Difficulty:

Yeah.

9

Compact $\implies$ Exhaustible by compacts

Added:

Mar 12, 2026

Difficulty:

Indeed.

13

Compact $\implies$ Strongly paracompact

Added:

Mar 12, 2026

Difficulty:

A subcover is a refinement. A finite subcover is star-finite.

14

Paracompact $\implies$ Metacompact

Added:

Mar 12, 2026

Difficulty:

If finitely many intersect a nbd around the point, finitely many will intersect the point.

15

Paracompact $\implies$ Countably paracompact

Added:

Mar 12, 2026

Difficulty:

If true for any covers, then true for countable ones.

16

Submetacompact $\implies$ Countably metacompact

Added:

Mar 12, 2026

Difficulty:

If true for any covers, then true for countable ones.

17

Countably compact $\implies$ Countably paracompact

Added:

Mar 12, 2026

Difficulty:

A subcover is a refinement. A finite subcover is star-finite.

18

Countably paracompact $\implies$ Countably metacompact

Added:

Mar 12, 2026

Difficulty:

This is Paracompact $\implies$ Metacompact, just countable this time.

25

Topological $n$ -manifold $\implies$ Locally $n$ -Euclidean

Added:

Mar 12, 2026

Difficulty:

By definition.

41

Has a dispersion point $\implies$ ¬ Empty

Added:

Mar 12, 2026

Difficulty:

If it has a dispersion point… it has… a point.

42

Discrete $\implies$ $T_1$

Added:

Mar 12, 2026

Difficulty:

Singletons are clopen.

52

(Totally disconnected ∧ Has multiple points) $\implies$ ¬ Connected

Added:

Mar 12, 2026

Difficulty:

The space is not a singleton.

67

Countable $\implies$ Cardinality $\lt\mathfrak c$

Added:

Mar 12, 2026

Difficulty:

This is obvious, so fun fact: The converse requires the continuum hypothesis.

68

Cardinality $\lt\mathfrak c$ $\implies$ Cardinality $\leq\mathfrak c$

Added:

Mar 12, 2026

Difficulty:

Big brain stuff.

74

Countable $\implies$ $\sigma$ -compact

Added:

Mar 12, 2026

Difficulty:

Singletons are compact.

75

(Injectively path connected ∧ Has multiple points) $\implies$ ¬ Cardinality $\lt\mathfrak c$

Added:

Mar 12, 2026

Difficulty:

The path between two distinct points has at least $\mathfrak c$ points.

80

(Functionally Hausdorff ∧ Has multiple points) $\implies$ ¬ Strongly connected

Added:

Mar 12, 2026

Difficulty:

The continuous map with $f(a) = 0$ and $f(b) = 1$ is not constant.

88

(Path connected ∧ Has multiple points) $\implies$ ¬ Totally path disconnected

Added:

Mar 12, 2026

Difficulty:

Take a path between two points. It’s not constant.

94

(Injectively path connected ∧ Has multiple points) $\implies$ ¬ Biconnected

Added:

Mar 12, 2026

Difficulty:

If $f$ continuous, $f([0, 1/3])$ and $f([2/3, 1])$ are connected.

IDs 1-99

Compact ⟹ \implies⟹ Countably compact

Countably compact ⟹ \implies⟹ Weakly countably compact

Countably compact ⟹ \implies⟹ Pseudocompact

Compact ⟹ \implies⟹ Locally relatively compact

Locally relatively compact ⟹ \implies⟹ Weakly locally compact

Exhaustible by compacts ⟹ \implies⟹ Weakly locally compact

Compact ⟹ \implies⟹ Exhaustible by compacts

Compact ⟹ \implies⟹ Strongly paracompact

Paracompact ⟹ \implies⟹ Metacompact

Paracompact ⟹ \implies⟹ Countably paracompact

Submetacompact ⟹ \implies⟹ Countably metacompact

Countably compact ⟹ \implies⟹ Countably paracompact

Countably paracompact ⟹ \implies⟹ Countably metacompact

Topological nnn-manifold ⟹ \implies⟹ Locally nnn-Euclidean

Has a dispersion point ⟹ \implies⟹ ¬ Empty

Discrete ⟹ \implies⟹ T1T_1T1​

(Totally disconnected ∧ Has multiple points) ⟹ \implies⟹ ¬ Connected

Countable ⟹ \implies⟹ Cardinality <c\lt\mathfrak c<c

Cardinality <c\lt\mathfrak c<c ⟹ \implies⟹ Cardinality ≤c\leq\mathfrak c≤c

Countable ⟹ \implies⟹ σ\sigmaσ-compact

(Injectively path connected ∧ Has multiple points) ⟹ \implies⟹ ¬ Cardinality <c\lt\mathfrak c<c

(Functionally Hausdorff ∧ Has multiple points) ⟹ \implies⟹ ¬ Strongly connected

(Path connected ∧ Has multiple points) ⟹ \implies⟹ ¬ Totally path disconnected

(Injectively path connected ∧ Has multiple points) ⟹ \implies⟹ ¬ Biconnected

Compact $\implies$ Countably compact

Countably compact $\implies$ Weakly countably compact

Countably compact $\implies$ Pseudocompact

Compact $\implies$ Locally relatively compact

Locally relatively compact $\implies$ Weakly locally compact

Exhaustible by compacts $\implies$ Weakly locally compact

Compact $\implies$ Exhaustible by compacts

Compact $\implies$ Strongly paracompact

Paracompact $\implies$ Metacompact

Paracompact $\implies$ Countably paracompact

Submetacompact $\implies$ Countably metacompact

Countably compact $\implies$ Countably paracompact

Countably paracompact $\implies$ Countably metacompact

Topological $n$ -manifold $\implies$ Locally $n$ -Euclidean

Has a dispersion point $\implies$ ¬ Empty

Discrete $\implies$ $T_1$

(Totally disconnected ∧ Has multiple points) $\implies$ ¬ Connected

Countable $\implies$ Cardinality $\lt\mathfrak c$

Cardinality $\lt\mathfrak c$ $\implies$ Cardinality $\leq\mathfrak c$

Countable $\implies$ $\sigma$ -compact

(Injectively path connected ∧ Has multiple points) $\implies$ ¬ Cardinality $\lt\mathfrak c$

(Functionally Hausdorff ∧ Has multiple points) $\implies$ ¬ Strongly connected

(Path connected ∧ Has multiple points) $\implies$ ¬ Totally path disconnected

(Injectively path connected ∧ Has multiple points) $\implies$ ¬ Biconnected