Functional analysis 1. Topological space

Expanding Analysis into General Topological Spaces Classical analysis and linear algebra extend naturally to settings where points live in abstract spaces. Convergent sequences, continuous maps, and linear operators are redefined beyond metrics to work in general topological spaces. The course centers on metric and normed spaces, with touches of topological vector spaces. Neighborhood-based and sequence-based viewpoints (Cauchy vs. Heine) are compared to see where equivalences persist or fail.

Open Sets Defined Axiomatically A topology on a nonempty set X is a family τ of subsets called open sets. It satisfies: ∅ and X are open; any union of open sets is open; any finite intersection of open sets is open. With τ fixed, (X, τ) is a topological space. This axiomatic approach abstracts familiar open intervals on the real line.

Multiple Topologies on the Same Set The same underlying set can carry several distinct topologies. Normed spaces illustrate this with the norm (strong) topology and the weaker Banach weak topology; dual spaces add the weak-star topology. Each induces different neighborhoods and thus different continuity and convergence behaviors. Keeping track of which topology is in play is essential.

Neighborhoods as Local Open Enclosures A neighborhood of a point x is any open set that contains x. Every open set containing x qualifies, so points typically have many neighborhoods. Because X itself is open, every point has at least one neighborhood. Notation like U(x), V(x), and W(x) indicates such local enclosures.

Extremal Topologies: Weakest and Strongest The weakest topology has only ∅ and X, giving each point just one neighborhood, X itself. The strongest topology is the full power set of X, where every subset is open and every set containing x is a neighborhood of x, even the singleton {x}. Intersections and unions behave trivially at these extremes. These cases bound all others by inclusion.

Comparing Topologies by Inclusion Topologies compare via inclusion: τ1 ⊆ τ2 means τ2 is stronger and τ1 is weaker. Inclusion defines a partial order on all topologies over X, so many pairs are incomparable. The strongest topology dominates all others, and the weakest is dominated by all. Later, norm vs. weak vs. weak-star topologies exemplify this hierarchy.

Closed Sets as Complements of Opens A subset A of X is closed if its complement X \ A is open. The family Φ of all closed sets mirrors τ via complements. Closed sets provide a dual language to work with the same structure. This duality often simplifies arguments by switching unions to intersections and vice versa.

Intersections and Finite Unions Preserve Closedness Any intersection of closed sets is closed, because complements turn intersections into unions of opens. Finite unions of closed sets are closed, since complements convert finite unions into finite intersections of opens. Both ∅ and X are closed. These properties are dual to the open-set axioms.

Describing a Topology via Its Closed Sets One can present a topology by specifying its closed sets instead of its opens. If a family Φ contains ∅ and X, is closed under arbitrary intersections and finite unions, then the complements of elements of Φ form a topology τ. In that topology, Φ is exactly the family of closed sets. This equivalence lets constructions start from either side.

Clopen Extremes and the Interest in Between In the strongest topology, every set is both open and closed. In the weakest topology, only ∅ and X enjoy that status. Real applications lie between these extremes, where openness and closedness carry information. The choice of topology shapes which sets and maps behave well.

Bases and Prebases for Constructing Topologies To build topologies efficiently from simple ingredients, bases and prebases are introduced. A base generates all open sets by unions; a prebase first generates a base via finite intersections. Detailed construction is deferred to keep focus on general properties. These tools enable concrete, algorithmic examples later.

Closure as Intersection of Closed Supersets The topological closure [S]τ is the intersection of all closed sets that contain S. Being an intersection of closed sets, it is itself closed. There is always something to intersect because X is closed and contains S. This closed-hull viewpoint parallels convex, linear, and conic hulls in linear spaces.

Closure Behavior in Weakest and Strongest Cases In the weakest topology, every nonempty set closes up to X. In the strongest topology, every set is already closed, so closure is identity. If S is closed, then [S]τ = S. These calibrate expectations for more nuanced topologies.

Adherence Points Characterized by Neighborhoods A point x is an adherence (contact) point of S if every neighborhood of x meets S. This definition uses only the neighborhood language. It captures the idea that S accumulates at x regardless of membership. Adherence points link closure with neighborhoods.

Closedness Equivalently Captures All Adherence Points A set S is closed if and only if it contains all its adherence points. If an adherence point lay outside S, its complement would give an open neighborhood disjoint from S, contradicting adherence. Conversely, if every exterior point admits a neighborhood missing S, the complement is open. This reframes closedness in purely local terms.

Openness via Local Neighborhoods Inside the Set A set A is open exactly when each x in A has a neighborhood U(x) contained in A. Then A equals the union of these neighborhoods and is open by the union axiom. This interior-point criterion complements the adherence characterization of closedness. It will be used repeatedly.

Closure Equals the Set of All Adherence Points The closure of S consists precisely of all adherence points of S. If x is in the closure, no neighborhood of x can avoid S; otherwise a closed superset excluding x would exist. Conversely, if every neighborhood of x meets S, x lies in every closed superset of S. Thus [S]τ is the set of all topological adherence points.

Sequences and Topological Convergence A sequence (xn) in X converges to x when every neighborhood U of x eventually contains xn. This generalizes the familiar ε–N notion from the real line. The topology induces the convergence concept without extra machinery. It unifies disparate convergence notions across spaces.

Convergence under Weakest and Strongest Topologies In the weakest topology, every sequence converges to every point because X is the only neighborhood. In the strongest topology, convergence forces eventual equality to the limit, since {x} is a neighborhood of x. Hence limit uniqueness is not guaranteed in arbitrary topologies. Separation axioms are needed to tame such behavior.

Stationary Tails Guarantee Convergence Any sequence that is eventually constant converges to that constant in any topology. This trivial class of convergent sequences always exists. In very strong topologies, it may be the only class of convergent sequences. It highlights how topology strength throttles convergence.

Topologizing Convergence in Function Spaces In spaces of functions, one often starts from a notion of convergence and then equips a topology that induces it. Doing so unlocks the machinery of topological vector spaces, revealing properties not visible from sequences alone. Distribution theory achieves this by clothing convergence in a vector topology in the sense of Laurent Schwartz. Rudin’s early chapters survey such topologies; classic texts on normed spaces prepare the ground. Many important topologies in analysis, including weak and weak-star, are not norm-induced.

Sequential Adherence and Sequential Closedness A point x is a sequential adherence point of S if there exists a sequence of points of S converging to x. A set is sequentially closed if it contains all such points. These notions depend on the topology-induced convergence but avoid direct neighborhood language. They mirror the classical Heine-style definitions.

Sequential Closure and Its Basic Properties The sequential closure of S is the set of all its sequential adherence points. It always contains S via constant sequences. Unlike topological closure, it need not be topologically closed. It will often be strictly smaller than the topological closure.

Sequential Implies Topological; Closures Compare Every sequential adherence point is a topological adherence point, so every topologically closed set is sequentially closed. Consequently, the sequential closure of S is contained in its topological closure. There may exist topological adherence points not realized as limits of sequences from S. This one-way implication underlies many examples.

The Zariski Topology on the Real Line On R, declare a set open precisely when its complement is at most countable. Equivalently, the closed sets are R together with all at-most-countable subsets. This topology is far weaker than the Euclidean one. It satisfies the axioms by dual closed-set properties.

Zariski Convergence Forces Eventual Constancy A sequence in R converges to x in the Zariski topology only if it is eventually equal to x. Any neighborhood of x excludes at most countably many points, so eventually all unequal terms fall outside the complement. The tail must sit inside every neighborhood, forcing stabilization at x. Thus the class of convergent sequences matches that of the strongest topology.

Every Subset Is Sequentially Closed in Zariski Because a sequence has only countably many distinct values, any sequential limit within Zariski must stabilize inside the originating set. Therefore every subset of R is sequentially closed. The sequential closure of any set equals the set itself. Sequential closedness becomes maximally permissive here.

Zariski Topological Closure Expands to R If S ⊂ R has the cardinality of the continuum and S ≠ R, its topological closure in the Zariski topology is R. Countable closed supersets cannot contain an uncountable S, leaving R as the only closed superset. Meanwhile the sequential closure remains S. This displays a strict gap between sequential and topological closures.

Sequential Closures Can Fail to Be Closed There exist spaces where a sequential closure is not sequentially closed. For real functions on [0,1] with pointwise convergence, sequentially closing C[0,1] yields the discontinuous functions of Baire class 1, not all discontinuous functions. A second sequential closure gives Baire class 2 and includes the Dirichlet function. Proving the Dirichlet function is not of class 1 uses Baire’s category theorem. Iterating never stabilizes, producing higher (even transfinite) Baire classes.

Building the Sequential Topology from Closedness The family of sequentially closed sets satisfies the axioms for closed sets of some topology. Define the sequential topology τseq by taking complements of these sets; τseq contains the original τ. Convergence of sequences under τseq coincides with convergence under τ. This construction canonically topologizes sequential closedness.

Discrete Sequential Topology and Non-Topologizable Limits In Zariski, every set is sequentially closed, so the induced sequential topology is discrete. Despite different open sets, the notion of sequence convergence is identical to the original. Starting from an abstract convergence (without a topology), the sequential topology it induces generally yields a weaker convergence. Some convergences are not topologizable at all; convergence almost everywhere for measurable functions is a classical example due to Luzin. This limits how far topology can encode purely sequential behavior.