Prove Corollary with all details ( use Topology discrete) Theorem 5.4: If Y is a connected subspace of a space X, then Ÿ is connected.Proof: Suppose Y is connected. For a change of pace, the connectedness of Ỹ willbe shown by proving that there is no continuous function from Ÿ onto a discrete two-point space.Consider a continuous function f: Ÿ → {a, b} from Ÿ into such a discretespace. We must show that f is not surjective. The restriction f | y cannot be surjective.This means that f maps Y to only one point of {a, b}, say a:KY) = {a}.%3DSince f is continuous, Theorem 4.11 guarantees thatKỸ) CRY) = {ā} = {a},so f is not surjective. Thus, by Theorem 5.1, Ý is connected.5.2 / Theorems on Connectedness137An examination of the preceding proof will reveal that Theorem 5.4 can bestrengthened as follows:Corollary: Let Y be a connected subspace of a space X and Z a subspace of Xsuch that Y C ZCỸ. Then Z is connected.

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connected subspace of a

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.2: Linear Independence, Basis, And Dimension

Problem 15EQ

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Question

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Prove Corollary with all details ( use Topology discrete)

Theorem 5.4: If Y is a connected subspace of a space X, then Ÿ is connected.
Proof: Suppose Y is connected. For a change of pace, the connectedness of Ỹ will
be shown by proving that there is no continuous function from Ÿ onto a discrete two-
point space.
Consider a continuous function f: Ÿ → {a, b} from Ÿ into such a discrete
space. We must show that f is not surjective. The restriction f | y cannot be surjective.
This means that f maps Y to only one point of {a, b}, say a:
KY) = {a}.
%3D
Since f is continuous, Theorem 4.11 guarantees that
KỸ) CRY) = {ā} = {a},
so f is not surjective. Thus, by Theorem 5.1, Ý is connected.
5.2 / Theorems on Connectedness
137
An examination of the preceding proof will reveal that Theorem 5.4 can be
strengthened as follows:
Corollary: Let Y be a connected subspace of a space X and Z a subspace of X
such that Y C ZCỸ. Then Z is connected.

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