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Theorem 8.3. The space Rstd is connected.

Theorem 8.3. The space Rstd is connected.

Advanced Engineering Mathematics
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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Could you explain how to show 8.3 in detail? 

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 AnB = 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 → Rgstd 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 and X itself. (6) For every pair of points p and q and every open cover {U«}ae1 0f X there exist a finite number of the Ua's, {Ua,, Ua,, Uaz».., Ug, } such that pE U«,, q E Uan; and for each i < n, Ua; n Ua41 + Ø. Theorem 8.3. The space Rstd is connected.
Transcribed Image Text: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 AnB = 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 → Rgstd 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 and X itself. (6) For every pair of points p and q and every open cover {U«}ae1 0f X there exist a finite number of the Ua's, {Ua,, Ua,, Uaz».., Ug, } such that pE U«,, q E Uan; and for each i < n, Ua; n Ua41 + Ø. Theorem 8.3. The space Rstd is connected.
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