5.1. General Topology

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5.1.1. Basic Notions

A topology T on a set X means we get a collection of subsets of X.

To consider some subsets of set X, they constitute set T, we usually take such operations between them:

take intersection of any finite number of elements of T;

take union of any set of elements of T. 

such operations are rational and viable, because when we verify if an element belongs to the intersection, we just have to validate a finite number of subsets; and when we verify if an element belongs to the union, we just need know if it belongs one subset.

anyway, the resulting set of such operations is a subset of X, it is our concern too, so let it belongs to T.

To get the completion of our object T, let empty set and X itself belongs to T.

When we concern such a structure T of set X, name (X, T) a topological space, and the elements of T are called open sets in X.

Why we name the element of T open set?