The topic is topology, but what even is a topology? This problem describes that

Question

The topic is topology, but what even is a topology? This problem describes that most fundamental of questions. Topology is concerned with the position of points in the space, the topology of the space is a way of defining closeness of points without assigning measurements. Let X be a set, what will be our topological, space, and a topology on X, Tau is a family of subsets of X such that: emptyset belongsto Tau and X belongsto Tau, the union of any elements of Tau is in Tau, and any finite intersection of element of Tau is in Tau. Historically the first person to use the term topology was Johann Listing. The elements of a topology are the open sets within X. The axioms of topology are inspired by work of Georg Cantor on open and closed sets. To get the hang of this definition lets verify the properties for a couple of examples. (a) Show that for any set X, the family Tau = {emptyset, X} is a topology. This topology is called the trivial topology. It is the smallest possible topology or the coarsest. At the other end of the spectrum would be the finest topology which is the power set of X, the set of all subsets.

Explanation / Answer

assume NULL=@=/fi and ^ = intersection.

for any set X, T = {@,X }

here T is the family and the upper and lower bound are @ and X respectively.

now according to the defination,

@ and X both belongs to T.

so, (@ U X) = X : X belongs to T.

and, (@ ^ X) =@. @ belongs to T.

here

these two elements map only each of them, so it is a one-to-one mapping.

so, this is a trivial topology.

On the other hand, the elements or sub sets of power set of X will map one to all and all to one basis.

so that will be the finest topology which is the set of all subsets. (proved)

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