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The components
A maximal connected sub set C of a topological space X is called a component
of the space thus C is component of X if and only if C is connected and C is not
properly contained in any larger connected sub space.
Remark: Singleton sets of a topological space are connected , it is evident that
component are non-empty.
Let X be connected , then X has only one component namely X itself.
Let X is discrete space ,then singleton sets are the only connected sub sets of X
and consequently they are maximal connected sets ,hence ,each singleton sets is
component for discrete space.
Theorem: every component of a topological space (X,T) is closed.
Proof: H.W
If X is a topological space then
1- Each point in X is contained in exactly one component of X.
2- The component of X form a partition of X that is any two component are
either disjoint or identical and the union of all the component is X.
3- Each connected subset of X is contained in a component of X.
4- Each connected subset of X which is both open and closed is a component
of X.
Proof: H.W