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I'm taking a course that uses the book "algebraic topology" by Allen Hatcher. I this book there are two different ways in which a pair (X,A) of a topological space X and a subspace A can be nice: They can have the "homotopy extension property" (HEP) and they can be a "good pair". The definition of HEP can be found on wikipedia, and we say that (X,A) is a good pair if A is closed and non-empty and is a deformation retract of a neightborhood in X.

Are there examples of good pairs that does not have the HEP? And are there pairs that have the HEP without being good pair? Are they equivalent under some assumption (e.g. X being $\mathbb{R}^n$?)

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There is a notion of neighborhood deformation retract pair, or NDR pair, (en.wikipedia.org/wiki/Deformation_retract). You can also see May's "Concise Course in Algebraic Topology". This is equivalent to possessing the HEP. It also is a little stronger than good pair. So, having HEP means good pair, but not vice versa. Can't think of a counterexample right now. –  Joe Johnson 126 Oct 21 '11 at 11:15

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