By Nicolas Bourbaki
Bourbaki Library of Congress Catalog #66-25377 revealed in France 1966
Read Online or Download Elements of mathematics. General topology. Part 1 PDF
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Extra info for Elements of mathematics. General topology. Part 1
Sample text
Note that the half-open interval [0, 1[, as a subset of the real line, is neither open nor closed. We now go ahead with our new equivalence result. 4 Equivalent subsets have equivalent closures. Proof Let S be a Euclidean set, and let X and Y be equivalent subsets of S. Suppose that f is a homeomorphism from S to itself sending X to Y . We will show that f sends X to Y . Take any s in X. We first show that f (s) belongs to Y , so we consider any neighbourhood N of f (s). Because f is continuous the pre-image M of N is a neighbourhood of s.
A little later we will show that ]0, ∞[, which is also homeomorphic to ]0, 1[, is not equivalent to any of X, Y, Z in the plane. 12 are non-equivalent in the sphere. 8 Again we consider subsets of the sphere, but here each subset is homeomorphic to an open disc. Let X be the sphere with its north pole removed and let Y be the southern hemisphere excluding the equator. The complement of X consists of the north pole, whereas the complement of Y is a hemisphere, so the complements of X and Y are certainly not homeomorphic.
Find six subsets of C each homeomorphic to ]0, 1[, no two equivalent in C. Show that no two of your subsets are equivalent in C (assuming that a cylinder is not homeomorphic to a disc). 4 Surfaces and Spaces In this chapter we give a descriptive account of surfaces, of which we have already met the plane, the sphere and the torus. There are many other surfaces, shortly to be described. The essential idea is that near each of its points a surface is just like the plane. 1 A Euclidean set S is a surface if each of its points has a neighbourhood homeomorphic to an open disc.