Topological Basis : Toplogy || Lecture 2

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This is Lecture 2 of our first course on Topology. In this lecture the instructor started from definition of basis for a topology ( Topological basis or basis of a topology). Then he gave two examples of topological basis ( basis for a topology ) .
Then he proved two important lemmas.
Main points of this lecture are as follows:
Definition of Basis for a Topology ( Topological Basis ) is as under:
The Basis for a Topology T on a set X is a set B which satisfies following two conditions:
1) every element of X is in some basis element (which an element of topological basis B)
2) if an element x of X is in the intersection of basis elements B1 and B2 then there is a basis element B3 in the intersection and contains x.

Lemma 1 is a natural extension of property 2 described above to n basis elements. Lemma 2 an alternate of property 2.

Further examples of topological basis will be described in the next lecture.
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#Toplogy #Basis