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Proof That Elements of a Connected Space Can be Joined By a Simple Chain Consisting of Elements of the Open Cover of the Space

Theorem

Letbe a connected set and letbe an open cover ofAny two points ofcan be joined by a simple chain consisting of elements of.

Proof

Subsetsofare said to form a simple chain between pointsandif

  1. Onlycontains

  2. Onlycontains

  3. if

Letand letbe the set of all points ofwhich are to be joined toby a simple chain consisting of elements of

Letthen there exists a simple chainfromtoAll elements of belong tosince the chainconnects x with any element ofhence

Sinceis open, so is

Suppose now thatSinceis a cover ofthere is an elementsuch that

Supposethen there is some chain fromtoand- a contradiction henceand

Henceis open andis closed.

We concudeand the theorem is proved.