analysis (differential/integral calculus, functional analysis, topology)

epsilontic analysis

infinitesimal analysis

computable analysis

Introduction

Basic concepts

triangle inequality

metric space, normed vector space

open ball, open subset, neighbourhood

metric topology

sequence, net

convergence, limit of a sequence

compactness, sequential compactness

differentiation, integration

topological vector space

Basic facts

continuous metric space valued function on compact metric space is uniformly continuous

…

Theorems

intermediate value theorem

extreme value theorem

Heine-Borel theorem

…

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

• open subset, closed subset, neighbourhood

• topological space, locale

• base for the topology, neighbourhood base

• finer/coarser topology

• closure, interior, boundary

• separation, sobriety

• continuous function, homeomorphism

• uniformly continuous function

• embedding

• open map, closed map

• sequence, net, sub-net, filter

• convergence

• category Top

  • convenient category of topological spaces

Universal constructions

• initial topology, final topology

• subspace, quotient space,

• fiber space, space attachment

• product space, disjoint union space

• mapping cylinder, mapping cocylinder

• mapping cone, mapping cocone

• mapping telescope

• colimits of normal spaces

Extra stuff, structure, properties

• nice topological space

• metric space, metric topology, metrisable space

• Kolmogorov space, Hausdorff space, regular space, normal space

• sober space

• compact space, proper map

  sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact

• compactly generated space

• second-countable space, first-countable space

• contractible space, locally contractible space

• connected space, locally connected space

• simply-connected space, locally simply-connected space

• cell complex, CW-complex

• pointed space

• topological vector space, Banach space, Hilbert space

• topological group

• topological vector bundle, topological K-theory

• topological manifold

Examples

• empty space, point space

• discrete space, codiscrete space

• Sierpinski space

• order topology, specialization topology, Scott topology

• Euclidean space

  • real line, plane

• cylinder, cone

• sphere, ball

• circle, torus, annulus, Moebius strip

• polytope, polyhedron

• projective space (real, complex)

• classifying space

• configuration space

• path, loop

• mapping spaces: compact-open topology, topology of uniform convergence

  • loop space, path space

• Zariski topology

• Cantor space, Mandelbrot space

• Peano curve

• line with two origins, long line, Sorgenfrey line

• K-topology, Dowker space

• Warsaw circle, Hawaiian earring space

Basic statements

• Hausdorff spaces are sober

• schemes are sober

• continuous images of compact spaces are compact

• closed subspaces of compact Hausdorff spaces are equivalently compact subspaces

• open subspaces of compact Hausdorff spaces are locally compact

• quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff

• compact spaces equivalently have converging subnet of every net

  • Lebesgue number lemma

  • sequentially compact metric spaces are equivalently compact metric spaces

  • compact spaces equivalently have converging subnet of every net

  • sequentially compact metric spaces are totally bounded

• continuous metric space valued function on compact metric space is uniformly continuous

• paracompact Hausdorff spaces are normal

• paracompact Hausdorff spaces equivalently admit subordinate partitions of unity

• closed injections are embeddings

• proper maps to locally compact spaces are closed

• injective proper maps to locally compact spaces are equivalently the closed embeddings

• locally compact and sigma-compact spaces are paracompact

• locally compact and second-countable spaces are sigma-compact

• second-countable regular spaces are paracompact

• CW-complexes are paracompact Hausdorff spaces

Theorems

• Urysohn's lemma

• Tietze extension theorem

• Tychonoff theorem

• tube lemma

• Michael's theorem

• Brouwer's fixed point theorem

• topological invariance of dimension

• Jordan curve theorem

Analysis Theorems

• Heine-Borel theorem

• intermediate value theorem

• extreme value theorem

topological homotopy theory

• left homotopy, right homotopy

• homotopy equivalence, deformation retract

• fundamental group, covering space

• fundamental theorem of covering spaces

• homotopy group

• weak homotopy equivalence

• Whitehead's theorem

• Freudenthal suspension theorem

• nerve theorem

• homotopy extension property, Hurewicz cofibration

• cofiber sequence

• Strøm model category

• classical model structure on topological spaces