Question

Show that the set S={{x}│x∈X} being a sub base generates a discrete topology on any set X

Answer 1

Answer on Question #42211 – Math - Topology

Show that the set $S = \{\{x\} | x \in X\}$ being a sub base generates a discrete topology on any set $X$.

Solution.

A subbase $S$ generates a basis $B$ for the topology $T$. A basis $B$ consisting of all finite intersections of elements of $S$, together with the set $X$ and the empty set. So a basis $B$ consisting of the set $S$, set $X$ and the empty set. Any subset of $X$ can be written as a union of all sets that consisting of a single element from initial subset (this sets are elements of $B$). Hence, the topology $T$, generated by a basis $B$, consisting of all subsets of $X$ and the topology $T$ is discrete.

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