> Letr: S S be a reflection of the circle (e.g. (x,y) (-x, y) in the...

Transcribed Image Text:

> Letr: S S be a reflection of the circle (e.g. (x,y) (-x, y) in the plane). The Klein bottle K is the quotient space of [0, 1] x S under the following equivalence relation: (0, 2)~ (1, r(2)) for all z S, and (t, z) is not equivalent to anything except itself, for t = 0, 1. (a) Explain why K is compact. (b) Let C C K be (the image of) the circle 1 x S, and let C C K be a small embedded circle inside (11) S. There is a continuous map g K R such that glk-c, and g/k-c, are injective. Assuming g exists as described, use Urysohn's Lemma to construct a continuous map of K into R = R x R, which is an imbedding. > Letr: S S be a reflection of the circle (e.g. (x,y) (-x, y) in the plane). The Klein bottle K is the quotient space of [0, 1] x S under the following equivalence relation: (0, 2)~ (1, r(2)) for all z S, and (t, z) is not equivalent to anything except itself, for t = 0, 1. (a) Explain why K is compact. (b) Let C C K be (the image of) the circle 1 x S, and let C C K be a small embedded circle inside (11) S. There is a continuous map g K R such that glk-c, and g/k-c, are injective. Assuming g exists as described, use Urysohn's Lemma to construct a continuous map of K into R = R x R, which is an imbedding.

Answer rating: 100% (QA)

The detailed ... View the full answer