Let $X = \{1,2,3,4, 5\}$ and $\mathcal{A}= \{ A : A \subset X, \#A=4 \}$. Show that the topology generated by $\mathcal{A}$ on $X$ is the discrete one.
So $\mathcal{A}$ consists of subsets of $X$ with $4$ elements each. I know that we can get a basis from a subbase by taking finite intersections of the elements of the subbase and that if the basis elements are singletons, then the topology on $X$ would be the discrete one. I have trouble expressing the basis as the intersections of elements of $\mathcal{A}$, how can I approach this?
Best Answer
A subbasis for a topology on $X$ is a collection of subsets of $X$ whose union equals $X$, hence $\mathcal{A}$ is a subbasis for a given topology. The topology generated by the subbasis $\mathcal{A}$ is defined to be the collection $\tau$ of all unions of finite intersections of elements of $\mathcal{A}$.
Pick all finite intersections of the elements in $\mathcal{A}$: you get for sure the singletons $\{1\}, \{2\}, \{3\}, \{4\}, \{5\} $, where for example you get $\{1\}$ as intersection of all the $4-$element sets that contain the element $1$. Hence the basis you built from the subbasis contains the singletons, hence the generated topology is the discrete one.