MATH527_HW12 - Serge Ballif MATH 527 Homework 12 December...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Serge Ballif MATH 527 Homework 12 December 7, 2007 Problem 1. Show that S n is not homeomorphic to any proper subspace of itself. Recall invariance of domain: Let U be open in R n (or we could use S n ). Let f : U R n be continuous and injective. Then f ( U ) is open in R n and f is an imbedding. Let U be a subspace of S n that is homeomorphic to S n via a homeomorphism h (i.e. U = f ( S n )). Then U is compact as the continuous image of the compact space S n . Since S n is Hausdorff, U is closed. However, by invariance of domain, U is open. S n is connected, so the only subsets of S n that are both open and closed are S n and . Therefore U = S n . Therefore, there is no homeomorphism from S n to a proper subspace of itself. Problem 2. Let A be a closed subspace of R n homeomorphic to R k . Compute the homology groups of R n - A . e H i ( R n - A ) = Z for i = n - k - 1, and e H i ( R n - A ) = 0 otherwise. To see this, we note that since A is closed, but not compact, it must be unbounded. Hence,
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 04/01/2008 for the course MATH 527 taught by Professor Rotman,regina during the Fall '07 term at Pennsylvania State University, University Park.

Page1 / 2

MATH527_HW12 - Serge Ballif MATH 527 Homework 12 December...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online