Question #217084

Establish a.necessary and sufficient condition for a.family of subsets of a set X to be a.Q base for a topology on X


1
Expert's answer
2021-07-29T11:40:02-0400

A family of subsets(Q) is said to be a base for a topology on x ifFor each xX, there exists BQ such that xBAlso if xB1B2 for some B1,B2Q, then there exists BQ such thatxBB1B2\text{A family of subsets(Q) is said to be a base for a topology on x if}\\\text{For each $x \in X$, there exists $B \in Q$ such that $x \in B$}\\\text{Also if $x \in B_1 \cap B_2$ for some $B_1,B_2 \in Q$, then there exists $B \in Q$ such that}\\x\in B \subseteq B_1 \cap B_2


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