Topology/Connectedness

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Contents

MotivationEdit

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.

ExamplesEdit

  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 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:
     

    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.

DefinitionsEdit

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 connectednessEdit

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

Proof:
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.

Proof:
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.

Proof:
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 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.

ExercisesEdit

  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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