Homework Answers
Given x1 6= x2; choose xi 2 (ai; bi) possibly (a1; b1) \ (a2;
b2) 6= ;:
We may assume that a2 < b1: If there is c 2 (a1; b1)\(a2; b2)
then try (a1; c)
and (c; b2). Otherwise try (a1; b1) and (a2; b2):
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(3E) Slinky Line. Consider R with the usual topology. Prove that R/N (see 3.12(e)) is Hausdorff...
Consider R with the usual Euclidean topology and let I = [0, 1] be the closed...
Consider R with the usual Euclidean topology and let I = [0, 1] be the closed unit interval of R with the subspace topology. Define an equivalence relation on R by r ~y if x, y E I and [x] = {x} if x € R – I, where [æ] denotes the equivalence class of x. Let R/I denote the quotient space of equivalence classes, with the quotient topology. Is R/I Hausdorff? Is so, prove so from the definition of...
Prove that in R^n with the usual topology, if a set is closed and bounded then it is compact.
Prove that in R^n with the usual topology, if a set is closed and bounded then it is compact.
2. Prove the following Theorems: (a). Prove that the real line with the standard topology is...
2. Prove the following Theorems: (a). Prove that the real line with the standard topology is Hausdorff. (b). Prove that int(ANB) = int(A) n int(B) Y is a homeomorphism. Then if X is a (c). If X and Y are topological spaces and f: X Hausdorff space then Y is Hausdorff. (d). Theorem 4.2
Only part (a). denotes the power set of the irrationals. (51) Michael's Line. Let S denote...
Only part (a).
denotes the power set of the irrationals.
(51) Michael's Line. Let S denote R with the topology generated by the base P(P) U {(a,b) b, a, b, E R} (see 2D(9)). Note that T(R) C T(S) where T(R) is the usual topology on R. a < (a) Show S is a completely Hausdorff space (use 5.29(a)) (b) Show S has a base of clopen sets and conclude that S is Tychonoff (see the comment before 5.13 Theorem...
3. (a) Let (R, τe) be the usual topology on R. Find the limit point set...
3. (a) Let (R, τe) be the usual topology on R. Find the limit point set of the following subsets of R (i) A = { n+1 n : n ∈ N} (ii) B = (0, 1] (iii) C = {x : x ∈ (0, 1), x is a rational number (b) Let X denote the indiscrete topology. Find the limit point set A 0 of any subset A of X. (c) Prove that a subset D of X is...
Consider the set A = {(x,0) ER|XE R} in R2 with the vertical line topology. Find...
Consider the set A = {(x,0) ER|XE R} in R2 with the vertical line topology. Find the limit points of A. Consider the set A = {(x,0) ERP |x E R} in R2 with the vertical line topology. Find the boundary of A. Consider the set A = {(x,0) ERP|X E R} in R2 with the standard topology. Find the boundary of A.
topology Consider The The for and radius 1 in see R² with felloeding secrets delay) :...
topology
Consider The The for and radius 1 in see R² with felloeding secrets delay) : _ 14-442+ IV-vel doo (ny) = max { 10 -4 2/V-val} XELU, 4) and y = 1 U2 V22 a) Sketch open ball Billo centered al (1,1) both (R3 d.) and (IR² doo) prove That if u is open subset of (R², do it is also an open subset of (R² doo) @ Also -Prove that if u is open subset (R², do then...
2. Consider the relation E on Z defined by E n, m) n+ m is even} equivalence relation (a) Prove that E is an (b) Let n...
2. Consider the relation E on Z defined by E n, m) n+ m is even} equivalence relation (a) Prove that E is an (b) Let n E Z. Find [n]. equivalence relation in [N, the equivalence class of 3. We defined a relation on sets A B. Prove that this relation is an (In this view, countable sets the natural numbers under this equivalence relation). exactly those that are are
2. Consider the relation E on Z defined by...
Topology 3. Either prove or disprove each of the following statements: (a) If d and p...
Topology
3. Either prove or disprove each of the following statements: (a) If d and p map (X, d) X, then the identity topologically equivalent metrics (X, p) and its inverse are both continuous are two on (b) Any totally bounded metric space is compact. (c) The open interval (-r/2, n/2) is homeomorphic to R (d) If X and Y are homeomorphic metric spaces, then X is complete if and only if Y is complete (e) Let X and Y...
Problem 4. Let n E N. We consider the vector space R” (a) Prove that for...
Problem 4. Let n E N. We consider the vector space R” (a) Prove that for all X, Y CR”, if X IY then Span(X) 1 Span(Y). (b) Let X and Y be linearly independent subsets of R”. Prove that if X IY, then X UY is linearly independent. (C) Prove that every maximally pairwise orthogonal set of vectors in R” has n + 1 elements. Definition: Let V be a vector space and let U and W be subspaces...
Consider R with the usual Euclidean topology and let I = [0, 1] be the closed unit interval of R with the subspace topology. Define an equivalence relation on R by r ~y if x, y E I and [x] = {x} if x € R – I, where [æ] denotes the equivalence class of x. Let R/I denote the quotient space of equivalence classes, with the quotient topology. Is R/I Hausdorff? Is so, prove so from the definition of...
2. Prove the following Theorems: (a). Prove that the real line with the standard topology is Hausdorff. (b). Prove that int(ANB) = int(A) n int(B) Y is a homeomorphism. Then if X is a (c). If X and Y are topological spaces and f: X Hausdorff space then Y is Hausdorff. (d). Theorem 4.2
Only part (a).
denotes the power set of the irrationals.
(51) Michael's Line. Let S denote R with the topology generated by the base P(P) U {(a,b) b, a, b, E R} (see 2D(9)). Note that T(R) C T(S) where T(R) is the usual topology on R. a < (a) Show S is a completely Hausdorff space (use 5.29(a)) (b) Show S has a base of clopen sets and conclude that S is Tychonoff (see the comment before 5.13 Theorem...
Consider the set A = {(x,0) ER|XE R} in R2 with the vertical line topology. Find the limit points of A. Consider the set A = {(x,0) ERP |x E R} in R2 with the vertical line topology. Find the boundary of A. Consider the set A = {(x,0) ERP|X E R} in R2 with the standard topology. Find the boundary of A.
topology
Consider The The for and radius 1 in see R² with felloeding secrets delay) : _ 14-442+ IV-vel doo (ny) = max { 10 -4 2/V-val} XELU, 4) and y = 1 U2 V22 a) Sketch open ball Billo centered al (1,1) both (R3 d.) and (IR² doo) prove That if u is open subset of (R², do it is also an open subset of (R² doo) @ Also -Prove that if u is open subset (R², do then...
2. Consider the relation E on Z defined by E n, m) n+ m is even} equivalence relation (a) Prove that E is an (b) Let n E Z. Find [n]. equivalence relation in [N, the equivalence class of 3. We defined a relation on sets A B. Prove that this relation is an (In this view, countable sets the natural numbers under this equivalence relation). exactly those that are are
2. Consider the relation E on Z defined by...
Topology
3. Either prove or disprove each of the following statements: (a) If d and p map (X, d) X, then the identity topologically equivalent metrics (X, p) and its inverse are both continuous are two on (b) Any totally bounded metric space is compact. (c) The open interval (-r/2, n/2) is homeomorphic to R (d) If X and Y are homeomorphic metric spaces, then X is complete if and only if Y is complete (e) Let X and Y...
Problem 4. Let n E N. We consider the vector space R” (a) Prove that for all X, Y CR”, if X IY then Span(X) 1 Span(Y). (b) Let X and Y be linearly independent subsets of R”. Prove that if X IY, then X UY is linearly independent. (C) Prove that every maximally pairwise orthogonal set of vectors in R” has n + 1 elements. Definition: Let V be a vector space and let U and W be subspaces...
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