is a property that seeks to generalize the notion of closed subset and totally bounded subset of Euclidean space
the idea is that a compact space has no “punctures” nor “missing endpoints” (i.e. it includes all limiting values of points)
a set/space 𝑋 is compact if every open cover of 𝑋 has a finite subcover
a subset 𝑌 of a set/space 𝑋 is compact if every open cover of 𝑌 has a finite subcover

a set 𝑋 is sequentially compact if every sequence of points in 𝑋 has a convergent subsequence converging to a point in 𝑋

a set 𝑋 is weakly countably compact if every infinite subset of 𝑋 has a limit point in 𝑋
this property generalizes a property of compact spaces

a set 𝑋 is countably compact if every countable open cover has a finite subcover

Compactness - Other

for metric spaces; compactness, limit point compactness, and sequential compactness are ALL equivalent
for topological spaces; compactness, limit point compactness, and sequential compactness are NOT equivalent

Compactness - Examples & Non-Examples

the open interval (0,1) is not compact because it excludes the limiting values of 0 and 1
the closed interval [0,1] is compact
the space of rational numbers (ℚ) is not compact because it has infinitely many “punctures” corresponding to the irrational numbers
the space of real numbers (ℝ) is not compact because it excludes the two limiting values +∞ and -∞
the extended real number line would be compact because it contains both infinities
the entire Euclidean space of any dimension is not compact since it is not bounded
any subset (subspaces) of Euclidean space is compact if and only if it is both closed and bounded

a subset 𝑆 of a mathematical space (𝑋) is called a relatively compact subspace if its closure is compact

Compactness vs Relative Compactness

Compact	Relatively Compact	Example
❌	❌
❌	✔️
✔️	❌	take any neighborhood of the particular point of an infinite particular point space. The neighborhood itself may be compact but is not relatively compact because its closure is the whole space which is not compact
✔️	✔️