Chuangjie Xu 2013 (updated in February 2015)

\begin{code}

{-# OPTIONS --without-K #-}

module UsingIrrelevantFunext.Space.Space where

open import Preliminaries.SetsAndFunctions hiding (_+_)
open import Preliminaries.NaturalNumber
open import Preliminaries.Boolean
open import Preliminaries.Sequence
open import Continuity.UniformContinuity
open import UsingIrrelevantFunext.Irrelevance
open import UsingIrrelevantFunext.Space.Coverage

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Instead of working with whole category of sheaves over the above site, we
consider only those concrete sheaves which amount to the following C-spaces. To
topologize a set X, we use functions ₂ℕ → X, called probes, rather than subsets
of X, called open. Thus a topology on a set X, in our sense, is a set of maps
₂ℕ → X, called the probes, satisfying some conditions. We call it the C-topology.

\begin{code}

probe-axioms : (X : Set)  Subset(₂ℕ  X)  Set
probe-axioms X P =
    (∀(x : X)   α  x)  P)
  × (∀(t : ₂ℕ  ₂ℕ)  t  C  ∀(p : ₂ℕ  X)  p  P  p  t  P)
  × (∀(p : ₂ℕ  X)  (Σ \(n : )  ∀(s : ₂Fin n)  (p  (cons s))  P)  p  P)
  × (∀(p p' : ₂ℕ  X)  p  P  (∀(α : ₂ℕ)  [ p α  p' α ])  p'  P)

TopologyOn : Set  Set₁
TopologyOn X = Σ \(P : Subset(₂ℕ  X))  probe-axioms X P

Space : Set₁
Space = Σ \(X : Set)  TopologyOn X

U : Space  Set
U = pr₁

Probe : (X : Space)  Subset(₂ℕ  U X) 
Probe X = pr₁ (pr₂ X)

cond₀ : (X : Space) 
        ∀(x : U X)   α  x)  Probe X
cond₀ (_ , _ , c₀ , _) = c₀

cond₁ : (X : Space) 
        ∀(t : ₂ℕ  ₂ℕ)  t  C  ∀(p : ₂ℕ  U X)  p  Probe X 
        p  t  Probe X
cond₁ (_ , _ , _ , c₁ , _) = c₁

cond₂ : (X : Space) 
        ∀(p : ₂ℕ  U X)  (Σ \(n : )  ∀(s : ₂Fin n)  p  (cons s)  Probe X) 
        p  Probe X
cond₂ (_ , _ , _ , _ , c₂ , _) = c₂

cond₃ : (X : Space) 
        ∀(p p' : ₂ℕ  U X)  p  Probe X  (∀(α : ₂ℕ)  [ p α  p' α ])  p'  Probe X
cond₃ (_ , _ , _ , _ , _ , c₃) = c₃

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Then we define continuous functions between spaces and show that they do form a
category.

\begin{code}

continuous : (X Y : Space)  (U X  U Y)  Set
continuous X Y f =  p  p  Probe X  f  p  Probe Y

Map : Space  Space  Set
Map X Y = Σ \(f : U X  U Y)  continuous X Y f

id-is-continuous : ∀{X : Space}  continuous X X id
id-is-continuous p pinP = pinP

∘-preserves-continuity : (X Y Z : Space) 
    ∀(f : U X  U Y)  continuous X Y f 
    ∀(g : U Y  U Z)  continuous Y Z g 
    continuous X Z (g  f)
∘-preserves-continuity X Y Z f cf g cg p pP = cg (f  p) (cf p pP)

continuous-constant : (X Y : Space)  U Y  Map X Y
continuous-constant X Y y =  _  y) ,  _ _  cond₀ Y y)

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