Topology · Chapter 07

Connectedness

Being in one piece, said in open sets rather than in journeys: separations, connected subsets, what a continuous map does to them, paths, and the pieces a space falls into.

X = U \cup V

U \cap V = \emptyset

γ : [0, 1] \to X

γ (0) = x

γ (1) = y

[a, b]

f (X)

π_{0} (X)

Lessons

Build the foundation

01Connected spacesA space falls apart when it can be split into two open pieces, neither empty, sharing no point. It is connected when no such split exists, so it is all one piece.Read lesson →
02Disconnected spacesConnectedness is defined by the absence of something, so showing that a space is connected means ruling out every separation, while showing that it is disconnected means producing one. Producing one is the easy half, and this lesson is about doing it well: one set that is open and closed at once is enough, and the sets that do the splitting are often cuts at a point the space does not have.Read lesson →
03Connected subsetsA subset is connected when it is connected as a subspace, and as with compactness the test can be written with open sets of the surrounding space instead of traces. Two facts then do most of the work in this chapter: connected pieces sharing a point unite into a connected whole, and the connected subsets of the line are exactly the intervals.Read lesson →
04Continuous images of connected spacesA continuous map cannot tear a space. Send a connected space onto another and the target is connected too, because a split in the target would pull back to a split in the source.Read lesson →
05Path connectednessThere is a second sense in which a space can be one piece: any two of its points can be joined by a path inside it. A path is a continuous map from a closed interval into the space, so this is a statement about journeys rather than about splittings. Path connectedness implies connectedness, and the two are not the same — but for the spaces met in this course they usually agree, and paths are far easier to produce.

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06Connected componentsA space that is not connected still falls into pieces that are, and the pieces can be named: the component of a point is the largest connected subset containing it. Every point lies in exactly one component, so the components cut the space up without overlap, and how many there are is a topological invariant — often the quickest way to show two spaces are not homeomorphic.Read lesson →

07Chapter practiceTwelve questions across the chapter, with no teaching blocks. Connectedness is shown by transport and disconnectedness by example, and knowing which of the two a question is asking for is half of answering it.Read lesson →

Mapping the lesson.

Connectedness

Build the foundation