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A useful result is the following proposition which tells us that the restriction of a continuous map is a continuous map. Its routine proof is left to the reader – see also Exercises 5.1 #8. The topological space (X, τ ) is said to be locally homeomorphic to (Y, τ 1 ) if there exists a local homeomorphism of (X, τ ) into (Y, τ 1 ).

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We shall apply Proposition 2.3.4, but first we need to show that B is a basis for some topology on R2 . To do this we show that B satisfies the conditions of Proposition 2.2.8. While is not an open set in R with the euclidean topology. (See Exercises 2.1 #1.) So B is a basis for some topology but not a basis for the euclidean topology on R. Suppose that a and b exist with the required property and show that this leads to a contradiction, that is something which is false.

  • So we begin with the definition of a fliter and find examples.
  • Is called the indiscrete topology and (X, τ ) is said to be an indiscrete space.
  • For example, a set S in a metric space is closed if and only if every convergent sequence of points in S converges to a point in S.
  • Similarly e is a limit point of A even though it is not in A.
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§A3.0 Introduction In this Appendix we give but a is school a waste of time taste of dynamical systems and chaos theory. Most of the material is covered by way of exercises. Some parts of this Appendix require some knowledge of calculus.

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If (X, τ ) is a countable discrete space, prove that it is homeomorphic to a subspace of the Hilbert cube. Verify that C, with the metric d described in Example 6.1.5, is homeomorphic to a subspace of the Hilbert cube. Prove that the function d from the product space (X, τ ) × (X, τ ) into R is continuous. Using show that if the metrizable space (X, τ ) is connected and X has at least 2 points, then X has the uncountable number of points. If (X, τ ) and (Y, τ 1 ) are path-connected spaces, prove that the product space (X, τ ) × (Y, τ 1 ) is path-connected. Observation will be of greater significance in the next section when we proceed to a discussion of products of an infinite number of topological spaces.


Indeed it provides a necessary condition for a subset S of C to be dense, for F equal to R or C, and X a compact Hausdorff space. Let (S, τ 1 ) be a subspace of the topological space (X, τ ). Let (Y, τ 1 ) be a subspace of a topological space (X, τ ). We shall conclude this section with extension theorems; the first and most important is the Tietze Extension Theorem which is of interest in itself, but also ˘ ech compactification in the next section. We is useful in our study of the Stone–C shall prove various special cases of the Tietze Extension Theorem before stating it in full generality. Provides a sufficient condition for a topological space to be metrizable.

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Image of a subspace of the Cantor Space (G, τ ). Further, if (X, τ 1 ) is compact, then the subspace can be chosen to be closed in (G, τ ). Let (X, τ ) be an uncountable set with the discrete topology. In this section we give an application of topology to another branch of mathematics.

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But we will show in Exercises A3.7 #2 that the doubling map does indeed depend sensitively on initial conditions. The set of all periodic points of f is dense in the set X, and is transitive. Be taken as there is a number of inequivalent definitions of chaos in the literature as well as many writers who are vague about what they mean by chaos. Our definition is that used by Robert L. Devaney, with a modification resulting from the work of a group of Australian mathematicians, Banks et al. , in 1992. However, R can be replaced by any closed interval .

Having seen that the topology of a metric space can be described in terms of convergent sequences, we should not be surprised that continuous functions can also be so described. If (X, τ ) and (Y, τ 1 ) are homeomorphic topological spaces, verify that (X, τ ) is locally homeomorphic to (Y, τ 1 ). To prove two topological spaces are homeomorphic we have to find a homeomorphism between them. But, to prove that two topological spaces are not homeomorphic is often much harder as we have to show that no homeomorphism exists.

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While an undergraduate at the University of Amsterdam he proved original results on continuous motions in four dimensional space. Defined axiomatically what today is called a Banach space. Now we shall define the sum and product of two ordinals α and β. Every finite product of countable sets is countable. So S × T is a countably infinite union of countably infinite sets and is therefore countably infinite. Our next proposition gives us a necessary condition for a subset S of C to be dense, namely that it separates points of .