Exercise 2.4.2 (Base for a discrete topology)

Question

Let $X$ be a topological space with the discrete topology. Find a base $\mathscr{B}$ of open sets for $X$ such that $\mathscr{B}$ is included in any other base of open sets for $X$.

Elu Thingol · Accepted Answer

Let $\mathscr{B}$ denote the set of all singletons $\{x\}$ for $x \in X$. First, $\mathscr{B}$ is a base for the discrete topology on $X$ since every subset of $X$ is a union of its singletons. Furthermore, if $\mathscr{B}^{\prime}$ is another base for $X$, then it has to contain the singletons since the singletons cannot be formed by taking unions over any sets other than themselves. In other words, $\mathscr{B}^{\prime}$ has to include $\mathscr{B}$.

Exercise 2.4.2 (Base for a discrete topology)

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Exercise 2.4.2 (Base for a discrete topology)

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