Theorem 3.3. Suppose X is a set and B is a collection of subsets of X. Then B is a basis for some topology on X if and only if (1) each point of X is in some element of B, and (2) if U and V are sets in B and p is a point in U n V, there is a set W in B such that pEW c (Un V).

Theorem 3.3. Suppose X is a set and B is a collection of subsets of X. Then B is a basis for some topology on X if and only if (1) each point of X is in some element of B, and (2) if U and V are sets in B and p is a point in U n V, there is a set W in B such that pEW c (Un V).

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Confidence Intervals

When we calculate intervals in probability and statistics, it can be simply termed as finding out a confidence interval. The confidence interval is usually calculated from the data set which is observed. The value of the parameter that needs to be calculated is present here only.

Question

Could you explain how to show 3.3 in detail? Thank you!

**Theorem 3.3.** Suppose $ X $ is a set and $ \mathcal{B} $ is a collection of subsets of $ X $. Then $ \mathcal{B} $ is a basis for some topology on $ X $ if and only if

1. Each point of $ X $ is in some element of $ \mathcal{B} $, and

2. If $ U $ and $ V $ are sets in $ \mathcal{B} $ and $ p $ is a point in $ U \cap V $, there is a set $ W $ in $ \mathcal{B} $ such that $ p \in W \subseteq (U \cap V) $.

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