Separation and connectedness · Definition

Connected component

topologyconnectednesstopological spaces

Before you start you need to know

It breaks a space into its maximal connected pieces and counts how many separate pieces it has.

Definition

The connected component \(C(x)\) of a point \(x\) of a space \(X\) is the largest connected subset \(A\subseteq X\) containing \(x\).

\[C(x)=\bigcup\{A\subseteq X : x\in A,\ A\text{ connected}\}\]

A union of connected sets sharing a point is connected; hence the union of all connected sets containing \(x\) is connected and as large as possible. This guarantees that \(C(x)\) exists and is unique. Moreover \(C(x)\) is always closed, since the closure of a connected set is connected.

Two components are equal or disjoint, so the components partition \(X\): \(x\) and \(y\) lie in the same component exactly when some connected set contains both. A space is connected when it has a single component, and it is discrete when the components are points and each point is open.

Components need not be open. In \(\mathbb{Q}\) the components are the points, yet no point is open; \(\mathbb{Q}\) is called totally disconnected without being discrete. If there are only finitely many components, however, each one is open and closed. In \(\mathbb{R}\setminus\{0\}\) there are two components, \((-\infty,0)\) and \((0,\infty)\).

A continuous function maps each component into one component of the target. Hence a homeomorphism maps components onto components and the number of components distinguishes spaces: \([0,1]\) and \([0,1]\cup[2,3]\) are not homeomorphic.

Think about it

Why can two distinct connected components not share a point?

Because their union would be connected and strictly contain each, contradicting that each is the largest connected set.

How many components does \(\mathbb{R}\setminus\{0\}\) have, and why is \(\mathbb{R}\) not homeomorphic to it?

Two; \(\mathbb{R}\) is connected with a single component, and a homeomorphism preserves the number of components.

Exercises 10

  1. In \(X=\{1,2,3,4\}\) the open sets are the unions of \(\{1,2\},\{3\},\{4\}\). What is the connected component of \(1\)?
  2. How many connected components does \(\mathbb{R}\setminus\{0,1\}\) have with the usual topology?
  3. How many connected components does a discrete space with 5 points have?
  4. In \(\mathbb{Q}\) with the usual topology, what is the connected component of \(0\)?
  5. Which statement about the connected components of a space is true?
  6. In \(X=\{1,2,3\}\) with open sets \(\varnothing,\{1\},\{2,3\},X\), what is the connected component of \(2\)?
  7. Among the 29 topologies on \(\{1,2,3\}\), how many have exactly 3 connected components?
  8. Among the 29 topologies on \(\{1,2,3\}\), how many have exactly 2 connected components?
  9. Let \(f:X\to Y\) be continuous and \(C\) a connected component of \(X\) containing \(x\). Where does \(f(C)\) lie?
  10. A space has finitely many connected components. Which statement is true?

References

  • James R. Munkres, Topology, 2nd edition, Pearson, 2000. §25
  • Stephen Willard, General Topology, Dover reprint, Dover Publications, 2004. cap. 6