William J. Pervin

Algebra of Sets

The algebra of sets defines the properties and laws of sets; the set-theoretic operations of union, intersection, and complementation; and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions and performing calculations, involving these operations and relations. The relation consisting of all ordered pairs is called the cartesian product. This chapter describes two types of elements called maximal element and minimal element. The natural numbers with their usual ordering are an example of an order-complete set. The rational numbers ordered by size are not order-complete; however, the real numbers are order-complete. A set with a linear ordering is called a chain. The family of subsets of a set, ordered by inclusion, is an example of a partially ordered set that is not a linearly ordered set.

Separation and Countability Axioms

A topological space X is a T 0 -space if the closures of distinct points are distinct. A topological space X is a T 1 -space if every subset consisting of exactly one point is closed. Therefore, a T 1 -space X is countably compact if every countable family of closed sets having the finite intersection property has a nonempty intersection. In a T 1 -space, the intersection of a monotone decreasing countable collection of nonempty, closed sets, at least one of that is countably compact, is nonempty. The property of a space being a T 1 -space is preserved by one-to-one, onto, open maps and, hence, is a topological property. Every finite T 1 -space has the discrete topology. In a T 1 -space the intersection of a monotone decreasing sequence of nonempty, closed sets, at least one of which is countably compact, is nonempty. A T 1 -space is countably compact if every infinite open covering has a proper subcover.

Function and Quotient Spaces

This chapter describes the function and quotient spaces. It explains the concept of compact – open topology or the topology of compact convergence. The topology of uniform convergence on compact is the same as the compact-open topology for the set of all bounded, continuous functions from the topological space to the metric space. As compact sets have many of the nice properties of finite sets, therefore, the topology induced by the family of all compact subsets is called the compact-open topology or the topology of compact convergence. Every pseudometric space is a completely normal first axiom space.

Complete Metric Spaces

The notion of a Cauchy sequence plays a very important role in the study of the real numbers and can be generalized to metric spaces immediately. A Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite number of elements of the sequence are less than that given distance from each other. The property of being a Cauchy sequence depends strongly on the particular metric for the space, in the sense that equivalent metrics cannot have the same Cauchy sequences. Every metric space is isometric to a dense subset of a complete metric space and all completions of a metric space are isometric. However, the completion of a metric space is separable if the space is separable. A metric space is complete if it is absolutely closed. A metric space is complete if the intersection of every nested sequence of nonempty closed balls with radii tending to zero is nonempty.

Cardinal and Ordinal Numbers

The equivalence relation of equipotence divides up any collection of sets into equivalence classes and the property that equipotent sets is called cardinal numbers. The cardinal numbers are a measure of the number of points in sets. Among the infinite sets, the denumerable numbers have the cardinal number. In set theory, an infinite set is a set that is not a finite set. The infinite sets can be countable or uncountable. A set that is equipotent to the set of natural numbers is called denumearble. A set that is either finite or denumerable is called countable. The set of all rational numbers is denumerable. The union of a denumerable number of denumerable sets is a denumerable set and every infinite set contains a denumerable subset. Therefore, every infinite set is equipotent to a proper subset of itself. The set of all real numbers is uncountable.

CHAPTER 8 – Product Spaces

Publisher Summary The topological product is, up to a homeomorphism, a well-defined, commutative, and associative operation. The product of two spaces (or a finite number of spaces) with a certain property should also have that property. Such properties are said to be invariant under finite products. The properties of being totally bounded, complete, and of the first category are countably productive for metric spaces. The family of all sets with finite complements is a filter in an infinite set and a filter can be generated by a family of sets with the finite intersection property. For metric spaces, the following properties are invariant under finite products: (1) boundedness, (2) total boundedness, and (3) completeness. Tichonov spaces can be topologically embedded in compact Hausdorff spaces in the product space of closed intervals.

CHAPTER 11 – Uniform Spaces

Publisher Summary A uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties, such as completeness, uniform continuity, and uniform convergence. A quasi-uniform space is quasi-metrizable if its quasi-uniformity has a countable base and every topological space is quasi-uniformizable. The interior of every member of a uniformity belongs to the uniformity so that every member of a uniformity is a neighborhood of the diagonal in the product topology induced by the uniformity. Furthermore, the family of closed symmetric members of the uniformity is also a base for the uniformity. Although every member of a uniformity is a neighborhood of the diagonal, the converse is not always true. In the usual metric uniformity for the reals, for example, not all neighborhoods of the diagonal belong to the uniformity.

CHAPTER 3 – Topological Spaces

Publisher Summary A topological space is a set of points, along with a set of neighbourhoods for each point, that satisfy a set of axioms relating points and neighbourhoods. The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts, such as continuity, connectedness, and convergence. Other spaces such as manifolds and metric spaces are the specializations of topological spaces with extra structures or constraints. The topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics. The branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology. The concept of a topological space is introduced in terms of the axioms for the open sets. The closure operator is just one of a number of operators that can be used to define a topology. To completely determine a topological space, both the points and the open sets in the space should be specified. The two topological spaces are the same if both the points and the family of open sets are the same in each.

CHAPTER 6 – Metric Spaces

Publisher Summary A metric space is a set where a notion of distance (called a metric) between the elements of the set is defined. The importance of a metric for a set is that it gives a very convenient way to define a topology for the set. Two metrics for a set are called equivalent if their induced topologies are the same. Even when a particular metrics is not important, the theorems for metric spaces are stated rather than for metrizable spaces. For spaces that are metrizable, the properties of limits and continuity assume a familiar form when rephrased into metric notation. Every metric space is a first axiom space. In the induced topology, every ball in a metric space is an open set and metrizability is a topological property. Every separable metric space is second axiom and every Lindelof metric space is the second axiom. A metric space is compact if it is sequentially compact.

Metrization and Paracompactness

According to Urysohns metrization theorem, every second axiom T 3 -space is metrizable. That is a compact Hausdorff space is metrizable if it has a countable base. This result is sometimes also called Urysohns second metrization theorem. A family of subsets of a topological space is called locally finite. The countable union of locally finite (discrete) families is called σ-locally finite. Every discrete family is locally finite but not conversely. In a compact space, every locally finite or discrete family of subsets is finite. For every open covering of a metric space, there is a locally finite open cover that refines it. Every regular Lindelof space is paracompact and every regular space that is either second axiom or σ-compact is paracompact. However, there are many unsolved problems concerning paracompact spaces. For example, it is not known whether the product of a paracompact space with the closed unit interval is even normal.