Theorem 8.11. For topological spaces X and Y, X ×Y is connected if and only if each of X and Y is connected.

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Theorem 8.11. For topological spaces X and Y, X ×Y is connected if and only if each of
X and Y is 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 and X itself.
(6) For every pair of points p and q and every open cover {Uq}aea Qf X there exist a finite
number of the Ua's, {U«,, U«,, Uaz,.., Ua, } such that p E Ug,, q E U,, and for each
i < n, Ua, n Uau1 # Ø.
Theorem 8.3. The space Rstd is connected.
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 (Ure, Ca) is
αελ
connected.
Theorem 8.6. Let C be a connected subset of the topological space X. If D is a subset of
X such that C cDC C, then D is connected.
Theorem 8.9. Let f : X
→ Y be a continuous, surjective function. If X is connected,
then Y is connected.
Theorem 8.10 (Intermediate Value Theorem). Let f : Rstd
map. If a, b e R and r is a point of R such that f(a) < r < f(b), then there exists a point
c in (a, b) such that f(c) =
Rstd be a continuous
= r.
Transcribed Image Text:Theorem 8.11. For topological spaces X and Y, X ×Y is connected if and only if each of X and Y is 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 and X itself. (6) For every pair of points p and q and every open cover {Uq}aea Qf X there exist a finite number of the Ua's, {U«,, U«,, Uaz,.., Ua, } such that p E Ug,, q E U,, and for each i < n, Ua, n Uau1 # Ø. Theorem 8.3. The space Rstd is connected. 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 (Ure, Ca) is αελ connected. Theorem 8.6. Let C be a connected subset of the topological space X. If D is a subset of X such that C cDC C, then D is connected. Theorem 8.9. Let f : X → Y be a continuous, surjective function. If X is connected, then Y is connected. Theorem 8.10 (Intermediate Value Theorem). Let f : Rstd map. If a, b e R and r is a point of R such that f(a) < r < f(b), then there exists a point c in (a, b) such that f(c) = Rstd be a continuous = r.
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