TopologyPlease don't reject saying not clear image. (i) True or False: If a path-connected space X is contractible (i.e. the identity mapid: X X. id(z) = r, is homotopic to a constant map c: X→ X, c(x) = 10), thenX is simply connected (i.e. 7(X, to) is trivial). The converse is not necessarily true.(ii) Assume X is path connected and 7(X, 2o) is abelian (ie. [f] lg = [g] * [f] for allF], Lg] E T1(X, Lo)). Show that for any yo € Y and any two paths aq 02 connecting1o to yo the homomorphisms Pı, P2 : 71(X.26) → TI(X, 30) with ở (/]) = [a,'fa]and 2(f]) = [a, fa) are the same homomorphism.E3(iii) Let p: X→X be a covering map and assume both X and X are path connected. Saya0, 1 X is a loop based at ro E X and assume go, o ep(ro}, jo 2o. If theunique lift a: (0, 1] X with a(0) = is such that a(1)= , then X is not simplyconnected.

As per Bartleby guidelines for more than one question asked only first should be answered. Please…

(i) True or False: If a path-connected space X is contractible (i.e. the identity map id : X X. id(r) = x, is homotopic to a constant map c: X→ X, c(a) = x0), then X is simply connected (i.e. T1(X, xo) is trivial). The converse is not necessarily true.

(i) True or False: If a path-connected space X is contractible (i.e. the identity map id : X X. id(r) = x, is homotopic to a constant map c: X→ X, c(a) = x0), then X is simply connected (i.e. T1(X, xo) is trivial). The converse is not necessarily true.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Confidence Intervals

When we calculate intervals in probability and statistics, it can be simply termed as finding out a confidence interval. The confidence interval is usually calculated from the data set which is observed. The value of the parameter that needs to be calculated is present here only.

Question

Topology Please don't reject saying not clear image.

(i) True or False: If a path-connected space X is contractible (i.e. the identity map
id: X X. id(z) = r, is homotopic to a constant map c: X→ X, c(x) = 10), then
X is simply connected (i.e. 7(X, to) is trivial). The converse is not necessarily true.
(ii) Assume X is path connected and 7(X, 2o) is abelian (ie. [f] lg = [g] * [f] for all
F], Lg] E T1(X, Lo)). Show that for any yo € Y and any two paths aq 02 connecting
1o to yo the homomorphisms Pı, P2 : 71(X.26) → TI(X, 30) with ở (/]) = [a,'fa]
and 2(f]) = [a, fa) are the same homomorphism.
E3
(iii) Let p: X→X be a covering map and assume both X and X are path connected. Say
a0, 1 X is a loop based at ro E X and assume go, o ep(ro}, jo 2o. If the
unique lift a: (0, 1] X with a(0) = is such that a(1)= , then X is not simply
connected.

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