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1 Let Sn: = {x € Rn+¹ |||| = 1} be the unit sphere in R+1, with the topology induced from the standard (Euclidean) topology on R+1, for n ≥ 1. Also let RP = S/ ~be the projective n-space, obtained as the quotient space of Sn by the equivalence relation ay if y = x or y = -x. (a) Show that S is path-connected, by constructing for any two points x, y € Sm an explicit path connecting them. (b) Show that Sn is locally path-connected. (c) Show that RP is path-connected and locally path-connected.

1 Let Sn: = {x € Rn+¹ |||| = 1} be the unit sphere in R+1, with the topology induced from the standard (Euclidean) topology on R+1, for n ≥ 1. Also let RP = S/ ~be the projective n-space, obtained as the quotient space of Sn by the equivalence relation ay if y = x or y = -x. (a) Show that S is path-connected, by constructing for any two points x, y € Sm an explicit path connecting them. (b) Show that Sn is locally path-connected. (c) Show that RP is path-connected and locally path-connected.

Advanced Engineering Mathematics
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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5. Let \( S^n = \{ x \in \mathbb{R}^{n+1} : \|x\| = 1 \} \) be the unit sphere in \( \mathbb{R}^{n+1} \), with the topology induced from the standard (Euclidean) topology on \( \mathbb{R}^{n+1} \), for \( n \geq 1 \). Also let \( \mathbb{RP}^n = S^n / \sim \) be the projective \( n \)-space, obtained as the quotient space of \( S^n \) by the equivalence relation \( x \sim y \) if \( y = x \) or \( y = -x \). (a) Show that \( S^n \) is path-connected, by constructing for any two points \( x, y \in S^n \) an explicit path connecting them. (b) Show that \( S^n \) is locally path-connected. (c) Show that \( \mathbb{RP}^n \) is path-connected and locally path-connected.
Transcribed Image Text:5. Let \( S^n = \{ x \in \mathbb{R}^{n+1} : \|x\| = 1 \} \) be the unit sphere in \( \mathbb{R}^{n+1} \), with the topology induced from the standard (Euclidean) topology on \( \mathbb{R}^{n+1} \), for \( n \geq 1 \). Also let \( \mathbb{RP}^n = S^n / \sim \) be the projective \( n \)-space, obtained as the quotient space of \( S^n \) by the equivalence relation \( x \sim y \) if \( y = x \) or \( y = -x \). (a) Show that \( S^n \) is path-connected, by constructing for any two points \( x, y \in S^n \) an explicit path connecting them. (b) Show that \( S^n \) is locally path-connected. (c) Show that \( \mathbb{RP}^n \) is path-connected and locally path-connected.
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