By S, Steenrod, N Eilenberg
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Extra resources for Foundations of Algebraic Topology
Example text
Any two disks on the plane are homeomorphic; a disk is homeomorphic to the inside of a square or triangle - the corresponding homeomorphisms are easy to construct directly. An interval (a;'b)- = {x E [J;£: a < x < b} on the real line [J;£ is homeomorphic to [J;£. From this it is evident that boundedness of a set and or its diameter are not topological invariants. This is not surprising: boundedness and diameter are definied in terms of a metric, and not in terms of a collection of open sets. A circle is not homeomorphic to a segment [a, b] = {x E [J;£: a ~ x ~ b} because any continuous map of a segment of itself has a fixed point, while a rotation of a circle through 90° about its center has no fixed point.
The map case qJ is a condensation). qJ is continuous if and only if f is continuous (in which Proposition 17. The map f is a quotient map if and only if qJ is a homeomorphism. In other words a quotient map can be characterized as a map whose image is canonically homeomorphic to the decomposition space it generates. Propositions 15 and 17 justify using the term "quotient space" to refer to a decomposition space of a topological space (and not just to the image of a topological space under a quotient map).
In order to address questions like Problems 1-5, it is important to have as wide as possible a spectrum of topological properties preserved by open (closed) maps. It is easy to show that the image under a continuous open map of a space satisfying the first axiom of countability is again a space satisfying the first axiom of countability. Open maps carry spaces with a countable base to spaces with a countable base. Proposition 7. If X is a Frechet-Uryson space and f: X --+ Y isa continuous closed map with f(X) = Y, then Y is also a Frechet-Uryson space.