4. A space X is called separable if X has a countable dense subset. Which of the following is notcorrect: *R with the usual topology is separableR with the left ray topology is separableO Rwith the co-countable topology is separableRwith the indiscrete topology is separableR with the cofinite topology is separable

4. A space X is called separable if X has a countable dense subset. Which of the following is not correct: R with the usual topology is separable O R with the left ray topology is separable O Rwith the co-countable topology is separable Rwith the indiscrete topology is separable O R with the cofinite topology is separable

4. A space X is called separable if X has a countable dense subset. Which of the following is not correct: R with the usual topology is separable O R with the left ray topology is separable O Rwith the co-countable topology is separable Rwith the indiscrete topology is separable O R with the cofinite topology is separable

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

See similar textbooks

Similar questions

4.4 Prove that if A is nowhere dense in discrete space (S, d) then A = Ø. 4.5 Prove that a closed set in the metric space (S, d) either is nowhere dense in S or else contains some nonempty open set.
2-4. Let X be a topological space and let A be a collection of subsets of X. Prove the following containments. (a) A≤Ã. AEA AEA (b) U 42 UĀ. AEA AEA (c) Int(n (₁) A) ≤ Int A. AEA (d) IntA2 U Int A. AEA AEA When A is a finite collection, show that equality holds in (b) and (c), but not necessarily in (a) or (d).
Let {V₁,..., Vn} be a spanning set for an inner product space V. Suppose that x and y are such that (x, vk) = (y, vk) for all k. Prove that x = y. . Please prove this from definitions and do not refer to previous review, hw, or gw problems (you will lose points if you do so). If you use a theorem or property from the text (that has been proved), please refer to which one.
Exercise 11. Recall the following Example of a topological space with the real line as underlying set (Example 26 in the notes): Consider on the real line R with the topology in which O C R is open if and only if for each x € O, there is r > x with [x, r) C O. This is easily checked to be a topology, called the Sorgenfrey line topology. (1) Given x < r in R, (a) Is [x, r) open? Justify why or why not. (b) Is [x, r) closed? Justify why or why not. (2) Is every open subset of the Sorgenfrey line also closed? Justify why or why not. (3) Is the property of Exercise 4 still valid when the topology on R is changed from the usual topology of R to the Sorgenfrey line topology? Explain. (4) Explain why the Sorgenfrey line is not homeomorphic to the real line with its usual topology.
Let x = (x1, x2, x3) ER³ and suppose you are given a subset UCR³ defined by U = {ER³ : x₂ < 5} Which of the following statements are necessarily true? (i) The complement of U contains a boundary point. (ii) The complement of U contains no interior points. (iii) U is not an open set. (iv) U is not a closed set. (A) (iii) only (B) (i) and (iv) (C) (ii) only (D) (i), (iii), and (iv) (E) (ii) and (iii) (F) (iv) only (G) (i) only (H) (iii) and (iv)
6. if x C y. (a) (b) Let R = {a: a is a cut}.Let x, y, R. We say that a
LHV IPITOR a space Y. (c) Let T be a cofinite topology topology on a set of real numbers and A=(3,6,8) show that whether A is a connected set or not A is compact set or not. G
1. Find the supremum, infimum, maximum, and minimum of each subset of R, if it exists. a. A = sup A max A inf A U[-1-] TEN min A = {":neN} b. B = sup B max B inf B min B c. C = {re Q:r² ≥ 1} sup C max C inf C min C Q Search DOLL 1