Answered: (3) Consider the topological space (R.… | bartleby

Elementary Linear Algebra (MindTap Course List)

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.CM: Cumulative Review

Problem 24CM

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(3) Consider the topological space (R, T), where
T = {U CR: U = 0 or R \ U is countable}.
(a) Show that T is a topology on R.
(b) Show that in (R, 7), any two non-empty open sets have non-empty
intersection.

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Determine whether the set R2 with the operations (x1,y1)+(x2,y2)=(x1x2,y1y2) and c(x1,y1)=(cx1,cy1) is a vector space. If it is, verify each vector space axiom; if it is not, state all vector space axioms that fail.
True or False: Consider the subsets (0, 1) and (2, 3) of R, the set of all realnumbers with the euclidean topology. The open intervals (0, 1) and (2, 3) are homeomorphic.
1) let R be a set of real number and T={Ø, R, A₁, aeR}, where A=(a,0). Show that whether T is a topology on R or not? 2) Let f be a function from a set X in to a topological space (Y,T) and T = {uCX, u=f'(v), VET). Show that T' is a topology on X.
let (x,t) be a topological space prove that (x,t) is not connected if and only if there exist A,B belongs to t with x= A union B and A intersect B = zero
Q1:A) Let X = R and r be the usual topology on R.If Y-[4,6). 1) Determine the open ,elosed ,open and closed for the following sets: (4,5), (4}. (5,6]. 2) Find 31
2. Let X be a topological space and let CX be the cone on X. (a) Show that CX is contractible. (b) Show that CX is locally connected if and only if X is locally connected.
Consider the set F=(-infinity, 1) U (9,infinity) as a subset of R with d(x,y)=abs(x-y). Show f is not compact by creating an open cover of F that has no finite subcover.
Consider the set F=(-infinity, 1) U (9,infinity) as a subset of R with d(x,y)=abs(x-y). Is F closed?
3A Given the Euclidean inner product space (R³(R),+,,*), where for each_x=(X1,X2,X3), y=(yı,y2,y3) ER³, x*y = 2(x₁y1+x3y3) + X2(Y₁+Y3) + (x1+x3)y2 + 3x2y2. Check whether the matrix A of the inner product (*) is squared with respect to the natural basis of R³, without computing its eigenvalues and eigenvectors.
Q2: a) Let f be a map from a set X into a space (Y,T) and T' {Us XaV e T,U f"(V).show that T is smallest topology on X,makes f continuous. b) Let T be a co finite topology on a set of reai numbers R and f: R →R be a map such that f(x) =k + 2/, show that whether f is T-T continuous or not.
Consider V = {(2,y) : x, y ≤ R}. Let and ⋆ be operations on this set defined in the following way: (2*1,31) © (2”, Y2) (2x1+x², Y₁ + Y2 − 1) k⋆ (2ª, y) = (2kx, ky − k +1) for k € R. - Show that the set V equipped with operations and is a real vector space. =
(1) Let (X, 7) be a topological space and let A CX. Let TA= {UNA: U ET}. (a) Show that TA is a topology on A. (b) Show that if 7 is Hausdorff, then T|A is Hausdorff.

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