Alphabeta Math

Topology

60 pages in 7 parts

Topology here is developed twice over, once with a distance and once without. Metric spaces come first: open sets, closure, convergence, continuity, completeness and the completion of a space, uniform continuity, and compactness in its equivalent metric forms, by covers, by sequences and by total boundedness. Dropping the distance and keeping the open sets gives the topological space, where continuity is a condition on preimages and subspaces, products, disjoint unions and quotients are initial or final topologies. Connectedness and compactness follow in that generality. Nets and filters repair what sequences no longer detect, the separation axioms grade how far apart a topology can hold points and closed sets, and Urysohn's lemma and Tietze's extension theorem produce the functions a normal space is supposed to have. Paracompactness and partitions of unity, the Tychonoff embedding, the Stone-Cech compactification, uniform spaces, the countability axioms and the metrization theorems follow. Function space topologies, Ascoli-Arzela and Stone-Weierstrass carry it into spaces of maps, and homotopy, the fundamental group and covering spaces open the algebraic side.

Much of the mathematics still to be built rests on it. Measure theory takes compactness, the Euclidean topology, Urysohn and Tietze, partitions of unity and the countability axioms. Functional analysis takes compactness, nets and filters and the function space topologies. Probability takes metric completeness and the Polish spaces. Algebraic topology begins at homotopy and the fundamental group. Differential geometry takes the product and quotient topologies and paracompactness. Commutative algebra puts the Zariski topology on a prime spectrum only after continuity and compactness exist.

Pathway

The parts run in order. Everything a page needs from this group has been read by the time you reach it, and the level on each row is how many dependency steps into the group that page sits.

  1. Part 1 · Metric spaces

    3 pages

    A first course in analysis uses the distance between two points without naming it, and three axioms are all it needs. Open sets, closure, convergence and continuity are defined from the metric here, then completeness gives the Cauchy criterion and the completion of a space, and uniform continuity separates a property of a function from a property at a point. Compactness in this setting has the three equivalent forms, by covers, by sequences and by total boundedness with completeness.

  2. Part 2 · Topological spaces

    4 pages · after Part 1

    Dropping the distance and keeping the open sets is what makes a topology, and continuity becomes preimages of open sets rather than an epsilon and a delta. Subspaces, products, disjoint unions and quotients are all initial or final topologies, so their characteristic properties turn continuity into a componentwise test. Connectedness and compactness are then stated for an arbitrary space, and the theorems they buy are the ones the metric versions had.

  3. Part 3 · Convergence and separation

    5 pages · after Part 2

    Sequences do not detect closure outside metric spaces, and nets and filters are the two repairs; ultrafilter convergence gives compactness a form with no covers in it. The separation axioms grade how far apart a topology can hold points and closed sets, from T0 to normality, with the Hausdorff condition equivalent to a closed diagonal. Urysohn's lemma produces the functions a normal space is supposed to have, Tietze extends them, and the last page asks which of these properties survive subspaces and products.

  4. Part 4 · Paracompactness, uniformities and compactification

    4 pages · after Part 3

    A locally finite refinement is what replaces a finite subcover when compactness is absent, and a partition of unity is what makes it usable for building functions. The Tychonoff embedding puts a completely regular space inside a cube of intervals, and the Stone-Cech compactification is the universal way to compactify it. A uniformity carries the comparison of pairs of points that a metric had and a topology does not, which is what completeness needs to be stated without numbers.

  5. Part 5 · Countability and metrization

    4 pages · after Parts 2, 3 and 4

    The Euclidean topology is where the product, the metric and every finite-dimensional norm agree, and it is the reference case for everything here. Countability axioms bound a space by how many open sets it takes to describe it, and the metrization theorems say when those bounds are enough to recover a metric: Urysohn for the second countable regular case, Nagata-Smirnov and Bing in general, Smirnov for the locally metrizable one. Complete metrizability and Cech-completeness carry Baire category to spaces with no metric attached.

  6. Part 6 · Function spaces

    3 pages · after Parts 2 and 4

    A set of functions carries no single topology, so this part puts four on the continuous maps between two spaces and says what each one measures, with the exponential law relating the compact-open topology to continuity of a map of two variables. Ascoli-Arzela describes the compact families through equicontinuity, and Stone-Weierstrass says when a family that separates points is uniformly dense.

  7. Part 7 · Homotopy and covering spaces

    7 pages · after Parts 4 and 5

    Homotopy deforms maps; based loops form the fundamental group. Covering maps lift paths and homotopies uniquely; universal-cover quotients classify connected coverings by conjugacy classes of subgroups, normal ones the regular coverings and normalizer quotients their deck groups; for the circle the cover is the line and the subgroups are those of Z. Seifert-van Kampen makes an open union's group a pushout, giving simply connected higher spheres, free groups for wedges of circles, the product formula and Z2 for the torus. Functoriality makes these obstructions: the circle is no retract of the disk, giving Brouwer's theorem and the fundamental theorem of algebra; Borsuk-Ulam constrains sphere-to-plane maps; topological-group loops commute; punctured spaces separate the plane from other Rn.