ONT Re: Topology
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Note 30
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| 3. Product and Quotient Spaces
|
| 3.1. Continuous Functions (cont.)
|
| A 'homeomorphism', or 'topological transformation',
| is a continuous one-to-one map of a topological space X
| onto a topological space Y such that f^(-1) is also continuous.
|
| If there exists a homeomorphism of one space onto another,
| the two spaces are said to be 'homeomorphic' and each
| is a 'homeomorph' of the other.
|
| The identity map of a topological space onto itself is always
| a homeomorphism, and the inverse of a homeomorphism is again
| a homeomorphism. It is also evident that the composition of
| two homeomorphisms is a homeomorphism. Consequently the
| collection of topological spaces can be divided into
| equivalence classes such that each topological space
| is homeomorphic to every member of its equivalence
| class and to these spaces only. Two topological
| spaces are 'topologically equivalent' iff they
| are homeomorphic.
|
| Two discrete spaces, X and Y, are homeomorphic iff there is a one-to-one
| function on X onto Y, that is, iff X and Y have the same cardinal number.
| This is true because every function on a discrete space is continuous,
| regardless of the topology of the range space. It is also true that
| two indiscrete spaces (the only open sets are the space and the void
| set) are homeomorphic iff there is a one-to-one map of one onto the
| other, because each function into an indiscrete space is continuous
| regardless of the topology of the domain space. In general, it may
| be quite difficult to discover whether two topological spaces are
| homeomorphic.
|
| The set of all real numbers, with the usual topology, is homeomorphic to the
| open interval (0, 1), with the relative topology, for the function whose
| value at a member x of (0, 1) is (2x-1) / x(x-1) is easily proved to be
| a homeomorphism. However, the interval (0, 1) is not homeomorphic to
| (0, 1) |_| (1, 2), for if f were a homeomorphism (or, in fact, just
| a continuous function) on (0, 1) with range (0, 1) |_| (1, 2),
| then f^(-1)[(0, 1)] would be a proper open and closed subset
| of (0, 1), and (0, 1) is connected.
|
| This little demonstration was achieved by noticing that one of the spaces is
| connected, the other is not, and the homeomorph of a connected space is again
| connected. A property which when possessed by a topological space is also
| possessed by each homeomorph is a 'topological invariant'. The proof that
| two spaces are not homeomorphic usually depends on exhibiting a topological
| invariant which is possessed by one but not by the other. A property which
| is defined in terms of the members of the space and the topology turns out,
| automatically, to be a topological invariant. Besides connectedness, the
| property of having a countable base for the topology, having a countable
| base for the neighborhood system of each point, being a T_1 space or
| being a Hausdorff space, are all topological invariants. Formally,
| topology is the study of topological invariants. *
|
| * A 'topologist' is a man who doesn't know the
| difference between a doughnut and a coffee cup.
|
| JLK, Gen Top, pages 87-88.
|
| John L. Kelley, 'General Topology',
| Van Nostrand Reinhold, New York, NY, 1955.
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