Compact space

A topological space $𝑋$ is called compact every open cover of $𝑋$ contains a finite subcover. #m/def/topology A subset is said to be compact iff it is such under the Subspace topology. A non-compact space may be made compact via a Compactification. A related notion is sequential compactness, which is equivalent in a second-countable space.

Complement characterisation

A topological space $𝑋$ is compact iff every family ${𝐹 𝛼} 𝛼 \in 𝐴$ of closed subsets such that $⋂ 𝛼 \in 𝐴 𝐹 𝛼 = \emptyset$ has a finite subfamily such that $⋂ 𝑛 𝑖 = 1 𝐹 𝛼 𝑖 = \emptyset$ .

Proof

Assume $𝑋$ is compact. Let ${𝐹 𝛼} 𝛼 \in 𝐴$ be a family of closed sets with $⋂ 𝛼 \in 𝐴 𝐹 𝛼 = \emptyset$ . Then ${𝑋 ∖ 𝐹 𝛼}$ is an open cover of $𝑋$ and therefore has a finite subcover ${𝑋 ∖ 𝐹 𝛼 𝑖} 𝑛 𝑖 = 1$ , in which case $⋂ 𝑛 𝑖 = 1 𝐹 𝛼 𝑖 = \emptyset$ .

For the converse, assume every family ${𝐹 𝛼} 𝛼 \in 𝐴$ of closed subsets of $𝑋$ such that $⋂ 𝛼 \in 𝐴 𝐹 𝛼 = \emptyset$ has a finite subfamily such that $⋂ 𝑛 𝑖 = 1 𝐹 𝛼 𝑖 = \emptyset$ . Let ${𝑈 𝛼} 𝛼 \in 𝐴$ be an open cover of $𝑋$ . Then $⋂ 𝛼 \in 𝐴 (𝑋 ∖ 𝑈 𝛼) = \emptyset$ so there exists a finite subfamily ${𝑋 ∖ 𝑈 𝛼 𝑖} 𝑛 𝑖 = 1$ such that $⋂ 𝛼 \in 𝐴 (𝑋 ∖ 𝑈 𝛼 𝑖) = \emptyset$ , in which case ${𝑈 𝛼 𝑖} 𝑛 𝑖 = 1$ is a finite subcover.

Properties

Other useful properties are limited to the Hausdorff-compact space.

Compactness is a stronger condition than Lindelöf
The continuous image of a compact space is compact
Compact subsets of a Hausdorff space are closed
Closed subsets of a compact space are compact
A continuous bijection from compact to Hausdorff is a homeomorphism

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