Could you explain how to show 3.16 in detail? Thank you! In particular, I am having hard time proving (<-) part. Theorem 3.16. Suppose X is a set and S is a collection of subsets of X. Then S is asubbasis for some topology on X if and only if every point of X is in some element of S.Definition. Let (X,J) be a topological space and let S be a collection of subsets of X.Then S is a subbasis for T if and only if the collection B of all finite intersections ofsets in S is a basis for T. An element of S is called a subbasis element or a subbasicopen set.Theorem 3.15. Let (X,J) be a topological space, and let S be a collection of subsets ofX. Then S is a subbasis for T if and only if(1) SCT, and(2) for each set U in T and point p in U there is a finite collection {V}{=1 of elements of Ssuch thatnPENVCU.i=1

Given that X is a set and S is a collection of subsets of X. Let B be the family of finite…

Answered: Theorem 3.16. Suppose X is a set and S…

Theorem 3.16. Suppose X is a set and S is a collection of subsets of X. Then S is a subbasis for some topology on X if and only if every point of X is in some element of S.

Elements Of Modern Algebra

8th Edition

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Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.1: Definition Of A Group

Problem 42E: 42. For an arbitrary set , the power set was defined in Section by , and addition in was...

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Numerical Integration

In a mathematical investigation, numerical integration comprises a wide group of calculations for computing the mathematical estimation of definite integral, and likewise, the term is moreover in some cases used to depict the numerical solution of differential equations. The most important aspect of this theory is error analysis.

Question

100%

Could you explain how to show 3.16 in detail? Thank you! In particular, I am having hard time proving (<-) part.

Theorem 3.16. Suppose X is a set and S is a collection of subsets of X. Then S is a
subbasis for some topology on X if and only if every point of X is in some element of S.
Definition. Let (X,J) be a topological space and let S be a collection of subsets of X.
Then S is a subbasis for T if and only if the collection B of all finite intersections of
sets in S is a basis for T. An element of S is called a subbasis element or a subbasic
open set.
Theorem 3.15. Let (X,J) be a topological space, and let S be a collection of subsets of
X. Then S is a subbasis for T if and only if
(1) SCT, and
(2) for each set U in T and point p in U there is a finite collection {V}{=1 of elements of S
such that
n
PENVCU.
i=1

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