nLab principle of enough functions

Context

Topology

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

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
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

Analysis

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

…

Constructivism, Realizability, Computability

constructive mathematics, realizability, computability

intuitionistic mathematics

propositions as types, proofs as programs, computational trinitarianism

Constructive mathematics

topos, homotopy topos
type theory, homotopy type theory
canonical form, univalence
Bishop set, h-set
decidable equality, decidable subset, inhabited set, subsingleton

Realizability

realizability topos
realizability model
realizability interpretation
effective topos
Kleene's first algebra, Kleene's second algebra
function realizability

Computability

computability
computation, computational type theory
computable function, partial recursive function
computable analysis, constructive analysis
computable set
persistent homology, effective homology
computable physics
Church-Turing thesis

Foundations

foundations

The basis of it all

mathematical logic
deduction system, natural deduction, sequent calculus, lambda-calculus, judgment
type theory, simple type theory, dependent type theory
collection, object, type, term, set, element
equality, judgmental equality, typal equality
universe, size issues
higher-order logic

Set theory

set theory

fundamentals of set theory
material set theory
presentations of set theory
structuralism in set theory
- material set theory
- structural set theory
  - categorical set theory
    - ETCS
    - structural ZFC
  - allegorical set theory
    - SEAR
class-set theory
constructive set theory
algebraic set theory

Foundational axioms

foundational axioms

basic constructions:
material axioms:
structural axioms:
- axiom of materialization
type theoretic axioms:
- axioms of set truncation
- homotopy type theoretic axioms:
  - univalence axiom
  - Whitehead's principle
axioms of choice:
large cardinal axioms:
strong axioms
- axiom of separation
- axiom of replacement
further
- reflection principle
axiom of tight apartness

Removing axioms

constructive mathematics
predicative mathematics

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Idea
Related concepts
References

Idea

The principle of enough functions states that the formal locale of the function space $\mathbb{R}^\mathbb{R}$ of continuous real endofunctions is a spatial locale.

The principle of enough functions is true in classical mathematics, but cannot be proven in neutral constructive mathematics and so has to be added as an axiom.

Related concepts

Heine-Borel theorem (The locale of the unit interval is a spatial locale.)
fan theorem (The locale of Cantor space is a spatial locale.)
monotone bar induction (The locale of the Baire space of sequences is a spatial locale.)

References

Ieke Moerdijk, Heine-Borel Does Not Imply the Fan Theorem, Journal of Symbolic Logic, vol. 49, no. 2, 1984, pp. 514–519. (doi:10.2307/2274182)

Created on April 14, 2025 at 21:27:45. See the history of this page for a list of all contributions to it.

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