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The components Definition: 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. Example: Let X be connected , then X has only one component namely X itself. Example: 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 Theorem: 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 I II