# Theory Discrete_Topology

theory Discrete_Topology
imports Analysis
```(*  Title:      HOL/Probability/Discrete_Topology.thy
Author:     Fabian Immler, TU München
*)

theory Discrete_Topology
imports "HOL-Analysis.Analysis"
begin

text ‹Copy of discrete types with discrete topology. This space is polish.›

typedef 'a discrete = "UNIV::'a set"
morphisms of_discrete discrete
..

instantiation discrete :: (type) metric_space
begin

definition dist_discrete :: "'a discrete ⇒ 'a discrete ⇒ real"
where "dist_discrete n m = (if n = m then 0 else 1)"

definition uniformity_discrete :: "('a discrete × 'a discrete) filter" where
"(uniformity::('a discrete × 'a discrete) filter) = (INF e:{0 <..}. principal {(x, y). dist x y < e})"

definition "open_discrete" :: "'a discrete set ⇒ bool" where
"(open::'a discrete set ⇒ bool) U ⟷ (∀x∈U. eventually (λ(x', y). x' = x ⟶ y ∈ U) uniformity)"

instance proof qed (auto simp: uniformity_discrete_def open_discrete_def dist_discrete_def intro: exI[where x=1])
end

lemma open_discrete: "open (S :: 'a discrete set)"
unfolding open_dist dist_discrete_def by (auto intro!: exI[of _ "1 / 2"])

instance discrete :: (type) complete_space
proof
fix X::"nat⇒'a discrete" assume "Cauchy X"
hence "∃n. ∀m≥n. X n = X m"
by (force simp: dist_discrete_def Cauchy_def split: if_split_asm dest:spec[where x=1])
then guess n ..
thus "convergent X"
by (intro convergentI[where L="X n"] tendstoI eventually_sequentiallyI[of n])
qed

instance discrete :: (countable) countable
proof
have "inj (λc::'a discrete. to_nat (of_discrete c))"
thus "∃f::'a discrete ⇒ nat. inj f" by blast
qed

instance discrete :: (countable) second_countable_topology
proof
let ?B = "range (λn::'a discrete. {n})"
have "⋀S. generate_topology ?B (⋃x∈S. {x})"
by (intro generate_topology_Union) (auto intro: generate_topology.intros)
then have "open = generate_topology ?B"
by (auto intro!: ext simp: open_discrete)
moreover have "countable ?B" by simp
ultimately show "∃B::'a discrete set set. countable B ∧ open = generate_topology B" by blast
qed

instance discrete :: (countable) polish_space ..

end
```