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Topological and Metric Spaces

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Topological and Metric Spaces Appendix A Topological and Metric Spaces This appendix will be devoted to the introduction of the basic proper- ties of metric, topological, and normed spaces. A metric space is a set where a notion of distance (called a metric) between elements of the set is defined. Every metric space is a topological space in a natural manner, and therefore all definitions and theorems about topological spaces also apply to all metric spaces. A normed space is a vector space with a special type of metric and thus is also a metric space. All of these spaces are generalizations of the standard Euclidean space, with an increasing degree of structure as we progress from topo- logical spaces to metric spaces and then to normed spaces. A.1 Sets In this book, we use classical notation for the union and intersection of sets. A\B denotes the difference of A and B,andAB is the symmetric difference. We use the notation A ⊂ B for strict inclusion, while A ⊆ B if A = B is allowed. Furthermore, An ↑ A means that An is a non decreasing sequence of sets and A = ∪An, while An ↓ A is a non increasing sequence of sets and A = ∩An.IfA and B are two sets, their Cartesian product A × B consists of all pairs (a, b), a ∈ A, and b ∈ B. If we have an infinite collection of sets, Aλ for λ ∈ Λ,then the Cartesian product of this collection is defined as: Aλ = f : Λ → Aλ : f(λ) ∈ Λ . λ λ The Axiom of Choice states that this product is nonempty as long as each Aλ is nonempty and Λ = ∅. H. Kunze et al., Fractal-Based Methods in Analysis, 357 DOI 10.1007/978-1-4614-1891-7, © Springer Science+Business Media, LLC 2012 358 A Topological and Metric Spaces A.2 Topological spaces The basic idea of a topological space is to try to find a very general setting in which the notions of convergence and continuity can be defined in some meaningful way. The key idea is to think of “open” sets as somehow measuring closeness. In this section, we give many of the definitions of the basic concepts in topology. As such, this section is primarily a list of definitions. A.2.1 Basic definitions Definition A.1. (Topology and topological space) A set X is a topo- logical space if a topology T on X has been chosen. A topology on X, T , is a family of subsets of X, called the open sets, that satisfies the following properties: • ∅ , X ∈T. ( • ∈T ∈T Ai for i =1, 2,...,n implies that i Ai . • ∈T ∈ ∈T Aλ for λ Λ implies λ Aλ . In words, a topology on X is a family of subsets that is closed under arbitrary unions and finite intersections. We call the complement of an open set a closed set. Sometimes it is more convenient to specify a collection smaller than the collection of all possible open sets. This is the idea behind the following definitions. Definition A.2. (Base and Subbase) A subfamily T0 ⊆T is called a base of T if each open set A ∈T can be written as a union of sets of T0.WealsosaythatT0 generates the topology T .Asubbase of a topology is a family of sets B such that their finite intersections form a base for the topology T ; that is, such that the collection n Ai : Ai ∈B i=1 is a base for T . If A ⊆ X,theinduced or subspace topology on A is given by inter- sections A ∩ O, for all O ∈T.Itiseasytoverifythatthiscollection satisfies the properties required for it to be a topology on A. A.2 Topological spaces 359 If X and Y are two topological spaces with topologies T1 and T2, respectively, then their Cartesian product X × Y is a topological space with the topology T generated by the base {A × B,A ∈T1,A ∈T2}. This topology is referred to as the product topology on X × Y.The product topology can also be defined for infinite products via the weak topology (see below). Definition A.3. (Closure, interior, boundary) Let E ⊆ X. ( • The set {C ⊆ X : E ⊆ C, C is closed} is called the closure of E and is denoted by cl E or E. • An open set A such that E ⊆ A is said to be a neighbourhood of E. • Apointx ∈ E is said to be an interior point of E if there exists a neighbourhood A ⊆ E of x. • The set of all interior points of E is called the interior ofE and denoted by int E. The interior of E is also equal to int E = {A ⊆ X : A ⊆ E,A is open }. • The boundary of E, ∂E, is the set cl E\ int E. If we have two topologies T1 and T2 on the same set X,wesaythat T1 is weaker than T2 (or T2 is stronger or finer than T1)ifT1 ⊆T2. It is easy to see that the intersection of any family of topologies is a topology. Furthermore, for any collection of topologies, there is a unique finest topology that is stronger than all of them. The finest topology of all is the discrete topology, in which every set is open. The coarsest topology is the trivial or indiscreet topology, in which the only open sets are the empty set and the whole space. A.2.2 Convergence, countability, and separation axioms Definition A.4. (Convergence) A sequence of points xn ∈ X, n ≥ 1, converges to x ∈ X as n → +∞ if every neighbourhood of x contains all xn with n ≥ n0 for some n0. A partially ordered set (Λ, () is called a directed system if for every α, β ∈ Λ thereisaγ ∈ Λ with α ( γ and β ( γ.Anet in X is a function x : Λ → X from a directed system to X. The net xλ converges to x ∈ X if for every neighbourhood N of x thereissomeγ ∈ Λ,soif λ ∈ Λ with γ ( λ,thenxλ ∈ N. 360 A Topological and Metric Spaces All sequences are nets, but not all nets are sequences. It is possible to show that a set C is closed if and only if it contains the limit for every convergent net of its points. In general, sequential convergence is not sufficient to define a given topology. However, if the space is first countable, then sequential convergence is sufficient to determine the topology (since each point has countably many sets to determine convergence and thus a net can be reduced to a sequence). Apointx ∈ E is said to be isolated if {x} is an open set. One consequence of this is that there is no sequence xn → x with xn = x for all n. Definition A.5. (First and second countable, separable) • A neighbourhood base at a point x ∈ X is a collection of sets N such that that for each neighbourhood U of x thereissomeN ∈N with x ∈ N ⊆ U. • A topological space X is first countable if each point has a countable neighbourhood base. • A topological space X is second countable if there is a countable base for the topology. • A topological space X is separable if there is a countable set {xn} such that that for any point x ∈ X and any neighbourhood U of x thereissomexn with xn ∈ U. In particular, this means that cl{xn : n ∈ N} = X. As an example, Rn with the usual notion of convergence (topology) is second countable. Every second countable space is first countable. Most of the “usual” spaces one encounters are first countable, or even second countable. Thus, for most situations sequential convergence is sufficient. Every second countable space is also separable. Conversely, any separable and first countable space is also second countable. Definition A.6. (Separation conditions) • A space is said to be T0 if for all distinct x, y ∈ X there is some open set U such that either x ∈ U and y/∈ U or x/∈ U and y ∈ U. • A space is said to be T1 if {x} is a closed set for all x ∈ X. • A space is said to be Hausdorff or T2 if for all distinct x, y ∈ X there are disjoint open sets U, V with x ∈ U and y ∈ V . These conditions are listed in order of increasing strength. That is, (Hausdorff ) T2 ⇒ T1 ⇒ T0. A.2 Topological spaces 361 A.2.3 Compactness The concept of compactness is central in many areas of topology. Roughly, it is a type of finiteness condition. Definition A.7. (Compactness) • A set K ⊆ X,iscompact if each open covering of K admits a finite ⊆ subcovering (i.e., whenever K λ Aλ for open sets Aλ,thereis { }⊆ ⊆ some finite subset λ1,λ2,λ3,...,λn Λ such that K j Aλj ). • If X itself is a compact set, then X is called a compact space. • A set E ⊆ X is said to be relatively compact if cl B is a compact set. • X is called locally compact if each point x ∈ X has a neighbourhood with compact closure. • If X can be represented as a countable union of compact sets, then it is said to be σ-compact. The set of all nonempty compact subsets of X is denoted by K(X). There is a standard construction that adds one point, located at “infinity”, to a locally compact space to obtain a compact space. The resulting space is called the one-point or Aleksandrov compactification. The corresponding open sets are open sets of X, and the added point ∞ has neighbourhoods that are complements to compact sets.
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