By A.V. Arkhangel'skii, L.S. Pontryagin, D.B. O'Shea, V.V. Fedorchuk
This is often the 1st of the encyclopaedia volumes dedicated to basic topology. It has components. the 1st outlines the fundamental ideas and structures of common topology, together with a number of themes that have no longer formerly been lined in English language texts. the second one half offers a survey of size idea, from the very beginnings to an important contemporary advancements. The relevant rules and strategies are taken care of intimately, and the most effects are supplied with sketches of proofs. The authors have suceeded admirably within the tricky activity of writing a publication so one can not just be available to the final scientist and the undergraduate, yet also will entice the pro mathematician. The authors' efforts to aspect the connection among extra really expert themes and the relevant topics of topology supply the ebook a extensive scholarly charm which a long way transcends slender disciplinary strains.
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Extra resources for General Topology I: Basic Concepts and Constructions Dimension Theory
Sample 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.