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Motivation[edit | edit source]

To best describe what is a connected space, we shall describe first what is a disconnected space. A disconnected space is a space that can be separated into two disjoint groups, or more formally:

A space is said to be disconnected iff a pair of disjoint, non-empty open subsets exists, such that .

A space that is not disconnected is said to be a connected space.

Examples[edit | edit source]

  1. A closed interval is connected. To show this, suppose that it was disconnected. Then there are two nonempty disjoint open sets and whose union is . Let be the set equal to or and which does not contain . Let . Since X does not contain b, s must be within the interval [a,b] and thus must be within either X or . If is within , then there is an open set within . If is not within , then is within , which is also open, and there is an open set within . Either case implies that is not the supremum.
  2. The topological space is disconnected:
    A picture to illustrate:
    The logo for this Wiki

    As you can see, the definition of a connected space is quite intuitive; when the space cannot be separated into (at least) two distinct subspaces.

Definitions[edit | edit source]

Definition 1.1

A subset of a topological space is said to be clopen if it is both closed and open.

Definition 1.2

A topological space X is said to be totally disconnected if every subset of X having more than one point is disconnected under the subspace topology

Theorems about connectedness[edit | edit source]

If and are homeomorphic spaces and if is connected, then is also connected.

Let be connected, and let be a homeomorphism. Assume that is disconnected. Then there exists two nonempty disjoint open sets and whose union is . As is continuous, and are open. As is surjective, they are nonempty and they are disjoint since and are disjoint. Moreover, , contradicting the fact that is connected. Thus, is connected.
Note: this shows that connectedness is a topological property.

If two connected sets have a nonempty intersection, then their union is connected.

Let and be two non-disjoint, connected sets. Let and be non-empty open sets such that . Let .
Without loss of generality, assume .

As is connected, ...(1).

As is non-empty, such that .

Hence, similarly, ...(2)
Now, consider . From (1) and (2), , and hence . As are arbitrary, is connected.

If two topological spaces are connected, then their product space is also connected.

Let X1 and X2 be two connected spaces. Suppose that there are two nonempty open disjoint sets A and B whose union is X1×X2. If for every x∈X, {x}×X2 is either completely within A or within B, then π1(A) and π1(B) are also open, and are thus disjoint and nonempty, whose union is X1, contradicting the fact that X1 is connected. Thus, there is an x∈X such that {x}×X2 contains elements of both A and B. Then π2(A∩{(x,y)}) and π2(B∩{(x,y)}), where y is any element of X2, are nonempty disjoint sets whose union is X2, and which are a union of open sets in {(x,y)} (by the definition of product topology), and are thus open. This implies that X2 is disconnected, a contradiction. Thus, X1×X2 is connected.

Exercises[edit | edit source]

  1. Show that a topological space is disconnected if and only if it has clopen sets other than and (Hint: Why is clopen?)
  2. Prove that if is continuous and surjective (not necessarily homeomorphic), and if is connected, then is connected.
  3. Prove the Intermediate Value Theorem: if is continuous, then for any between and , there exists a such that .
  4. Prove that is not homeomorphic to (hint: removing a single point from makes it disconnected).
  5. Prove that an uncountable set given the countable complement topology is connected (this space is what mathematicians call 'hyperconnected')
  6. a)Prove that the discrete topology on a set X is totally disconnected.

    b) Does the converse of a) hold (Hint: Even if the subspace topology on a subset of X is the discrete topology, this need not imply that the set has the discrete topology)

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