Topology · Chapter 06

Compactness

The property that thins every open cover: what it means for a subset, what it does to a continuous map, and what it amounts to in a metric space and in the line.

K \subseteq i = 1 ⋃ n U_{i}

U

[a, b]

f (K)

K \subseteq R^{n}

x_{n_{k}} \to x

B_{ε} (x)

K = K

Lessons

Build the foundation

01Open coversTo cover a space is to choose open sets that between them leave nothing out. Compactness asks whether any such choice, however large, can be cut down to finitely many that still cover.Read lesson →
02Compact spacesProving a space compact takes an argument about every cover. Denying it takes one cover and a reason it cannot be thinned, which is much less work and settles the matter.Read lesson →
03Compact subsetsCompactness was defined for a space, and a subset is called compact when it is compact in the topology it inherits. That reads as a statement about open sets of the subset; it can also be read as a statement about open sets of the space around it, and the two agree. The second reading is the one used in practice, because it needs no traces.Read lesson →
04Continuous images of compact spacesSend a compact space continuously onto another and the target is compact too. Cover the target, pull the cover back, thin it where compactness lives, and push the thinned cover forward.Read lesson →
05Compact subsets of metric spacesIn a metric space compactness forces two things at once: the set is closed, and it is bounded. Neither implication needs anything new — a finite subcover is produced and read. The converse is where the trouble starts: closed and bounded is not enough in a general metric space, and the line is special.Read lesson →

06Sequential compactnessThere is a second way to say that a space leaves no room to escape: every sequence in it has a subsequence that converges inside it. This is sequential compactness, stated in sequences rather than in covers. In a metric space the two properties coincide, which is why analysis can use either; in general spaces they part company.Read lesson →

07The Heine–Borel theoremIn the line and in every finite-dimensional real space, compactness has a description anyone can check: closed and bounded. That is the Heine–Borel theorem. It is a theorem about those spaces and not a definition of compactness — two lessons ago gave a closed bounded metric space that is not compact — and keeping the two apart is the point of this lesson.Read lesson →

08Chapter practiceEight questions across both properties, with no teaching blocks. Deciding which of the two a question is about, and which direction the argument runs, is the work here.Read lesson →

Mapping the lesson.

Compactness

Build the foundation