topology - Chapter 2 Topological Spaces A topological space...

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Unformatted text preview: Chapter 2 Topological Spaces A topological space ( X, τ ) is a set X with a topology τ , i.e., a collection of subsets of X with the following properties: 1. X ∈ τ , ∅ ∈ τ . 2. If A, B ∈ τ , then A ∩ B ∈ τ . 3. For any collection { A α } α , if all A α ∈ τ , then ∪ α A α ∈ τ . The sets in τ are called open sets , and their complements are called closed sets . A base of the topology τ is a collection of open sets such that every open set is a union of sets in the base. The coarsest topology has two open sets, the empty set and X , and is called the trivial topology (or indiscrete topology ). The finest topology contains all subsets as open sets, and is called the discrete topology . In a metric space ( X, d ) define the open ball as the set B ( x, r ) = { y ∈ X : d ( x, y ) < r } , where x ∈ X (the center of the ball), and r ∈ R , r > 0 (the radius of the ball). A subset of X which is the union of (finitely or infinitely many) open balls, is called an open set . Equivalently, a subset U of X is called open if, given any point x ∈ U , there exists a real number > 0 such that, for any point y ∈ X with d ( x, y ) < , y ∈ U . Any metric space is a topological space, the topology ( metric topology , topology induced by the metric d ) being the set of all open sets. The metric topology is always T 4 (see below a list of topological spaces). A topological space which can arise in this way from a metric space, is called a metrizable space . A quasi-pseudo-metric topology is a topology on X induced similarly by a quasi-semi-metric d on X , using the set of open d-balls B ( x, r ) as the base. In particular, quasi-metric topology and pseudo-metric topology are the terms used in Topology for the case of, respectively, quasi-metric and semi-metric d . In general, those topologies are not T . Given a topological space ( X, τ ), a neighborhood of a point x ∈ X is a set containing an open set which in turn contains x . The closure of a subset of a topological space is the smallest closed set which contains it. An open cover of X is a collection L of open sets, the union of which is X ; its subcover is a cover K such that every member of K is a member of L ; its refinement M.M. Deza and E. Deza, Encyclopedia of Distances , 5 9 DOI 10.1007/978-3-642-00234-2 2, c Springer-Verlag Berlin Heidelberg 2009 60 2 Topological Spaces is a cover K , where every member of K is a subset of some member of L . A collection of subsets of X is called locally finite if every point of X has a neighborhood which meets only finitely many of these subsets. A subset A ⊂ X is called dense if it has non-empty intersection with every non-empty open set or, equivalently, if the only closed set containing it is X ....
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topology - Chapter 2 Topological Spaces A topological space...

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