Topology
2nd Edition
ISBN: 9780134689517
Author: Munkres, James R.
Publisher: Pearson,
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Chapter 2.13, Problem 3E
Show that the collection
a topology on X?
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Chapter 2 Solutions
Topology
Ch. 2.13 - Prob. 1ECh. 2.13 - Consider the nine topologies on the set X=a,b,c...Ch. 2.13 - Show that the collection Tc given in Example 4 of...Ch. 2.13 - a If {T} is a family of topologies on X, show that...Ch. 2.13 - Show that if A is a basis for a topology on X,...Ch. 2.13 - Show that the topologies of l and K are not...Ch. 2.13 - Consider the following topologies on : T1 = the...Ch. 2.13 - a Apply Lemma 13.2 to show that the countable...Ch. 2.16 - Show that if Y is a subspace of X, and A is a...Ch. 2.16 - If and are topologies on X and is strictly...
Ch. 2.16 - Consider the set Y=[1,1] as a subspace of .Which...Ch. 2.16 - A map f:XY is said to be an open map if for every...Ch. 2.16 - Let X and X denote a single set in the topologies ...Ch. 2.16 - Exercises Show that the countable collection...Ch. 2.16 - Prob. 7ECh. 2.16 - Exercises If L is a straight line in the plane,...Ch. 2.16 - Exercises Show that the dictionary order topology...Ch. 2.16 - Exercises Let I=[0,1]. Compare the product...Ch. 2.17 - Let C be a collection of subsets of the set X....Ch. 2.17 - Show that if A is closed in Y and Y is closed in...Ch. 2.17 - Show that if A is closed in X and B is closed in...Ch. 2.17 - Show that if U is open in X and A is closed in X,...Ch. 2.17 - Let X be an ordered set in the order topology....Ch. 2.17 - Prob. 6.1ECh. 2.17 - Prob. 6.2ECh. 2.17 - Let A, B, and A denote subsets of a space X. Prove...Ch. 2.17 - Prob. 7ECh. 2.17 - Let A, B, and A denote subsets of a space X....Ch. 2.17 - Let A, B, and A denote subsets of a space X....Ch. 2.17 - Let A, B, and A denote subsets of a space X....Ch. 2.17 - Let AX and BY. Show that in the space XY, AB=AB.Ch. 2.17 - Show that every order topology is Hausdorff.Ch. 2.17 - Show that the product of two Hausdorff spaces is...Ch. 2.17 - Show that a subspace of a Hausdorff space is...Ch. 2.17 - Show that X is Hausdorff if and only if the...Ch. 2.17 - Prob. 14ECh. 2.17 - Show the T1 axiom is equivalent to the condition...Ch. 2.17 - Prob. 16.1ECh. 2.17 - Consider the five topologies on given in Exercise...Ch. 2.17 - Consider the lower limit topology on and the...Ch. 2.17 - Prob. 19ECh. 2.18 - Prove that for functions f:, the definition of...Ch. 2.18 - Prob. 2ECh. 2.18 - Let X and X denote a single set in the two...Ch. 2.18 - Let X and X denote a single set in the two...Ch. 2.18 - Given x0X and y0Y, show that the maps f:XXY and...Ch. 2.18 - Show that subspace (a,b) of is homeomorphic with...Ch. 2.18 - Prob. 6ECh. 2.18 - (a) Suppose that f: is continuous from the right,...Ch. 2.18 - Let Y be an ordered set in the order topology. Let...Ch. 2.18 - Let {A} be a collection of subsets of X; let X=A....Ch. 2.18 - Let f:AB and g:CD be continuous functions. Let us...Ch. 2.18 - Prob. 11ECh. 2.19 - Prove Theorem 19.2. Theorem 19.2. Suppose the...Ch. 2.19 - Prove Theorem 19.3. Theorem 19.3. Let A be a...Ch. 2.19 - Prove Theorem 19.4. Theorem 19.4. If each space X...Ch. 2.19 - Show that (X1Xn1)Xn is homeomorphic with X1Xn.Ch. 2.19 - One of the implications stated in Theorem 19.6...Ch. 2.19 - Let be the subset of consisting of all sequences...Ch. 2.19 - Given sequences (a1,a2,...) and (b1,b2,...) of...Ch. 2.19 - Show that the choice axiom is equivalent to the...
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Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.Similar questions
- Give an example of a relation R on a nonempty set A that is symmetric and transitive, but not reflexive.arrow_forwardProve that if f is a permutation on A, then (f1)1=f.arrow_forwardLabel each of the following statements as either true or false. The least upper bound of a nonempty set S is unique.arrow_forward
- Find mappings f,g and h of a set A into itself such that fg=hg and fh. Find mappings f,g and h of a set A into itself such that fg=fh and gh.arrow_forward23. Let be the equivalence relation on defined by if and only if there exists an element in such that .If , find , the equivalence class containing.arrow_forwardLabel each of the following statements as either true or false. Every upper bound of a nonempty set S must be an element of S.arrow_forward
- 13. Consider the set of all nonempty subsets of . Determine whether the given relation on is reflexive, symmetric or transitive. Justify your answers. a. if and only if is subset of . b. if and only if is a proper subset of . c. if and only if and have the same number of elements.arrow_forwardTrue or False Label each of the following statements as either true or false. If is an equivalence relation on a nonempty set, then the distinct equivalence classes of form a partition of.arrow_forwardLabel each of the following statements as either true or false. Let R be a relation on a nonempty set A that is symmetric and transitive. Since R is symmetric xRy implies yRx. Since R is transitive xRy and yRx implies xRx. Hence R is alsoreflexive and thus an equivalence relation on A.arrow_forward
- Let (A) be the power set of the nonempty set A, and let C denote a fixed subset of A. Define R on (A) by xRy if and only if xC=yC. Prove that R is an equivalence relation on (A).arrow_forwardTrue or False Label each of the following statements as either true or false. Let be an equivalence relation on a nonempty setand let and be in. If, then.arrow_forwardA relation R on a nonempty set A is called asymmetric if, for x and y in A, xRy implies yRx. Which of the relations in Exercise 2 areasymmetric? In each of the following parts, a relation R is defined on the set of all integers. Determine in each case whether or not R is reflexive, symmetric, or transitive. Justify your answers. a. xRy if and only if x=2y. b. xRy if and only if x=y. c. xRy if and only if y=xk for some k in . d. xRy if and only if xy. e. xRy if and only if xy. f. xRy if and only if x=|y|. g. xRy if and only if |x||y+1|. h. xRy if and only if xy i. xRy if and only if xy j. xRy if and only if |xy|=1. k. xRy if and only if |xy|1.arrow_forward
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