Alphabeta Math

Topology

64 pages in 7 parts

The topology pages cover both metric and general spaces. In metric spaces they treat convergence, completeness and completion, uniform continuity, compactness and discrete or countable bases. General topology develops open-set continuity, subspaces, products and quotients; connectedness, compactness and separation axioms; nets and filters; paracompactness, partitions of unity and metrization. Uniform spaces and compactifications extend these ideas, while function-space topology, Ascoli–Arzelà and Stone–Weierstrass study spaces of maps. The homotopy strand introduces the fundamental group, van Kampen, covering-space classification, applications and compact connected surfaces. The pages connect point-set methods with algebraic and geometric topics, while retaining tools beyond sequential convergence.

Real Analysis supplies convergence and compactness tools, while Set Theory supports choice and cardinal arguments; Combinatorics and Differential Geometry contribute specialized inputs. The resulting compactness, separation and covering-space methods support Algebraic Topology, Complex Analysis, Differential Geometry and Differential Topology. They also underpin Scheme Theory, Functional Analysis, Partial Differential Equations, Measure Theory and Probability as these subjects study spaces and maps.

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.

    • Topological Spaces and Continuity26 results

      Metric spaces provide the motivating examples: metric balls generate a topology, and the established metric notions of closure, convergence, and continuity supply comparison results.

      12 definitions, 4 lemmas, 5 theorems, 4 false statements, 1 remarkExamples & counterexamples
    • Subspaces, Products, and Quotients27 results

      Subspaces, products, disjoint unions and quotients are organised through initial and final topologies.

      7 definitions, 2 lemmas, 8 theorems, 1 corollary, 3 examples, 5 false statements, 1 remarkExamples & counterexamples
    • Connectedness29 results

      A space is connected when it cannot be split into two nonempty open pieces.

      6 definitions, 2 lemmas, 12 theorems, 2 corollaries, 1 example, 5 false statements, 1 remarkExamples & counterexamples
    • Compactness31 results

      Compactness is the hypothesis under which an infinite covering argument collapses to a finite one.

      5 definitions, 4 lemmas, 15 theorems, 1 corollary, 5 false statements, 1 remarkExamples & counterexamples

  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, and in a metric space, assuming choice, separated closed cores produce discrete open families at each scale whose countable union is a σ-discrete basis, with the conventions the metrization criteria use.

  5. Part 5 · Countability and metrization

    5 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

    8 pages · after Parts 4, 5 and 6

    Homotopy deforms maps; based loops form the fundamental group. Coverings lift paths and homotopies uniquely; subgroups up to conjugacy classify connected coverings, normal ones regular with normalizer-quotient deck groups; the circle's cover is the line; its subgroups are those of Z. Seifert-van Kampen makes open unions pushouts: simply connected higher spheres, free groups for wedges, product formulae, Z2 for the torus. Functoriality yields 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; punctures separate the plane from other Rn. Triangulation and polygon moves give every nonempty compact connected surface a normal form: orientability and Euler characteristic, 2−2g or 2−k, classify it.