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Suppose (X, T ) is a space, Y is a set, and f : X → Y is a map of setsthat is one-to-one and onto. Define f(T ) to be the set {f (U)|U ∈ T }. Call(Y, f(T )) the relabeling of (X, T ) by f.Proposition 3.5. The relabeling (Y, f(T )) is a topological space and therelabeling map f : (X, T ) → (Y, f(T )) is a homeomorphism.Proof. For any {Ui}i∈Ielements of T , we havef([i∈IUi) =[i∈If(Ui).This implies that f(T ) satisfies the axiom about unions is satisfied. For anyU, V ∈ T , we havef(U ∩ V ) = f(U) ∩ f(V )since f is one-to-one. So f (T ) satisfies the axiom about intersections. Fi-nally, since f is onto, f(X) = Y and Y ∈ f(T ). So (Y, f(T )) is a space.The map f is continuous since f−1(f(U)) = U for any U in T . ForU ∈ T , the p ul lback by f−1is f(U), which is clearly in f(T ). So f is ahomeomorphism.When we st ud y a topological space, we are not usually interested inspecific points, or in the names for specific points. The map f is giving anew name to each point of X, but is not otherwise changing the space.In fact, there is a converse to the observation the relabeling is a homeo-morphism:Proposition 3.6. Let f : (X, T ) → (Y, S) be a homeomorphism. ThenS = f(T ). In particular, (Y, S) is equal to the relabeling of (X, T ) by f .Proof. Suppose U ∈ S. Then since f is continuous, f−1(U) ∈ T . ThenU = f(f−1(U)) is in f(T ).Now suppose U ∈ f(T ). Then U = f(V ) for V ∈ T . Then since f−1is continuous with respect to S, f(V ) (the pullback by f−1(V )) is i n S. SoS = f(T ).To show that two spaces are homeomorphic, usually one has to constructa homeomorp hi sm between them. To show that spaces are not h ome omor -phic, one h as to show that the spaces differ with respect to some propertythat is preserved under homeomorphism. Since a homeomorphism is a rela-beling of the points in a space, any property that is defined using only thetopological st ru ct u r e of a space will be preserved under homeomorphism.9Properties not preserved under homeomorphisms are not part of th e topo-logical structure of a space, but come from s ome kind of additional structurethat we may have imposed separately: geometric structure, algebraic struc-ture, order structure, names of special points.Simple examples of topological properties of spaces include:• Existence of finite closed sets.• Existence of infinite closed sets.• Existence of finite open sets.Much of this course will be concerned with defining topological prop-erties, p r oving relations between them, and showing that certain examplesexhibit or fail to exhibit these properties.4 Closures, limit points and boundaries4.1 Closed setsRecall that a subset A of a space X is closed if and only if its complementX\A is open.There is a convenient characterization of t h e set of closed subsets of atopological space.Proposition 4.1. The closed sets of a topological space (X, T ) satisfy thefollowing axioms:1. Arbitrary intersections of closed sets are closed.2. Finite unions of closed sets are closed.3. The set X and the empty set ∅ are both closed.Conversely, if C is a set of subsets of X satisfying these axioms, then C isthe set of closed sets for some topology on X.The proof is an exercise using DeMorgan’s laws on set operations.Closed sets can be used to det ec t continuity.Proposition 4.2. A function between spaces f : X → Y is continuous ifand only if for every closed A ⊂ Y , the preimage f−1(A) ⊂ X is closed.10Proof. The preimages of complementary sets are complementary: if Y =A ∪ U , then X = f−1(A) ∪ f−1(U). So f pulls back closed sets to closedsets if and only if f pulls back open sets to open sets.Since infinite intersections of closed sets are closed, it is useful to makethe following defi ni t i on.Definition 4.3. Let X be a topological space and S a subset of X. Theclosure of S, denoted¯S, is the intersection of all closed sets containing S.Example 4.4. In the standard topology on R, the closure of a boundedinterval is the closed interval with the same endpoints.Some near-trivial observations about closed sets:Proposition 4.5. Let X be a space and S any subset of X.• The closure¯S is closed.• S is closed if and only if S =¯S.• The closure of the closure of S (write it S−−) is equal to¯S.Definition 4. 6. A subset S of a space X is dense if its closure is t he wholespace:¯S = X.Example 4.7. In the standard topology on R, t h e rationals and the irra-tionals are both dense.Existence of proper, dense subsets of differ ent sizes (finite or infinite) isan example of a topological property.4.2 Sequences and limit pointsFrom calculus and analysis, we are used to probing the t opological proper-ties of R using sequences. For completeness, we recall the definition of asequence.Definition 4.8. A sequence of points in a space X i s simply a f un ct i onfrom the natu ral numbers N to X. By convention, the funct i on inputs aredenoted as subscripts, so that we write a sequence as {pi}i∈Nfor pi∈ X.Naturally we must red efi n e convergence in a purely topological context.11Definition 4.9. A sequence {pi}i∈Nin a space X converges to a point p ∈ Xif for each open neighborhood U of p, there is a number N ∈ N (dep e nd i ngon U) such that for all i > N, we have pi∈ U.A lingui s t ic convenience: we say a set U contains a tail of the sequence{pi}i∈Nif there is some N ∈ N with all piin U for i > N . So a sequenceconverges to a point if every open neighborhood of the point contains a tailof the sequence.These definitions make the following important fact nearly a trivialty.Proposition 4.10. Suppose f : X → Y is continuous and {pi}i∈Nis asequence in X converging to p ∈ X. Then {f(pi)} converges to f(p).Proof. Let U be an open neighborhood of f(p) . Then f−1(U) is an openneighborhood of p. Then f−1(U) contains a tail of the sequence {pi}i∈N.Then U contains a tail of the sequence {f(pi)}i∈N.The topological space structure lets us tell when a point is “near” a set;such a point is called a limit point.Definition 4.11. Let X be a topological space and let S be a subset ofX. A point p in X is a limit point of S if every open neighb or hood U of pcontains a point of S other than p:U ∩ S\{p} 6= ∅.Example 4.12. In R with the usual topology, the limit points of an intervalare the endpoints of the interval. Note that it doe s not matter whether theseendpoints are contained in th e interval or not.Proposition 4.13. For any set S in any topological space X,


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CALTECH MA 109A - Lecture notes

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