Could you explain how to show 8.5 in detail? Theorem 8.5. Let {Ca}aea be a collection of connected subsets of X, and let E be anotherconnected subset of X such that for each a in 1, E n Ca # Ø. Then E U (Urea Ca) isαελconnected.Definition. Let X be a topological space. Then X is connected if and only if X is notthe union of two disjoint non-empty open sets.Definition. Let X be a topological space. Subsets A, B in X are separated if and onlyif An B = A n B = Ø. Thus B does not contain any limit points of A, and A does notcontain any limit points of B. The notation Xare separated sets.A | B means X = A U B and A and BTheorem 8.1. The following are equivalent:(1) X is connected.(2) There is no continuous function f : X → Rstd such that f(X) = {0,1}.(3) X is not the union of two disjoint non-empty separated sets.(4) X is not the union of two disjoint non-empty closed sets.(5) The only subsets of X that are both closed and open in X are the empty set andX itself.(6) For every pair of points p and q and every open cover {Uq}a€a 0f X there exist a finitenumber of the Ua's, {Ux,, U«,, Uaz,.., Ua, } such that p E Ua,, q E U,, and for eachi < n, Ua; n Uai+1# Ø.Theorem 8.3. The space Rstd is connected.

Answered: Image /qna-images/answer/dd9e9fb2-4645-428a-b2f2-1df180bf8f5a.jpg

Answered: Theorem 8.5. Let {Ca}gea be a…

Theorem 8.5. Let {Ca}gea be a collection of connected subsets ofX, and let E be another connected subset of X such that for each a in 1, E n Ca # Ø. Then E U (Uaea Ca) is соппеcted.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.4: Ordered Integral Domains

Problem 1E: Complete the proof of Theorem 5.30 by providing the following statements, where and are arbitrary...

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Could you explain how to show 8.5 in detail?

Theorem 8.5. Let {Ca}aea be a collection of connected subsets of X, and let E be another
connected subset of X such that for each a in 1, E n Ca # Ø. Then E U (Urea Ca) is
αελ
connected.
Definition. Let X be a topological space. Then X is connected if and only if X is not
the union of two disjoint non-empty open sets.
Definition. Let X be a topological space. Subsets A, B in X are separated if and only
if An B = A n B = Ø. Thus B does not contain any limit points of A, and A does not
contain any limit points of B. The notation X
are separated sets.
A | B means X = A U B and A and B
Theorem 8.1. The following are equivalent:
(1) X is connected.
(2) There is no continuous function f : X → Rstd such that f(X) = {0,1}.
(3) X is not the union of two disjoint non-empty separated sets.
(4) X is not the union of two disjoint non-empty closed sets.
(5) The only subsets of X that are both closed and open in X are the empty set andX itself.
(6) For every pair of points p and q and every open cover {Uq}a€a 0f X there exist a finite
number of the Ua's, {Ux,, U«,, Uaz,.., Ua, } such that p E Ua,, q E U,, and for each
i < n, Ua; n Uai+1
# Ø.
Theorem 8.3. The space Rstd is connected.

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