Topological Space

definition example

Let $X$ be a Set and $\mathcal{P}(X)$ its Power Set. A topology on $X$ is a subset $\mathrm{Op}\subseteq \mathcal{P}(X)$, whose elements are called open sets, such that

(a) whole set: $X\in \mathrm{Op}$; (b) binary intersections: $U,V\in \mathrm{Op}\Rightarrow U\cap V\in \mathrm{Op}$; (c) arbitrary unions: for any family $(U_i{)}_{i\in I}$ of opens, ${\bigcup }_{i\in I}{U}_i\in \mathrm{Op}$; with $I=\varnothing$ this says $\varnothing \in \mathrm{Op}$.

If $U={\bigcup }_{i\in I}{U}_i$ we say $(U_i{)}_{i\in I}$ covers $U$ (an open cover). A topological space is a pair $(X,\mathrm{Op})$. A continuous function $f:(X,{\mathrm{Op}}_X)\to (Y,{\mathrm{Op}}_Y)$ is a function with $f^{-1}(U)\in {\mathrm{Op}}_X$ for every $U\in {\mathrm{Op}}_Y$. Spaces and continuous maps form the category $\mathbf{Top}$.

Sources: 7 Sketches §7.3.2, Definition 7.25, Examples 7.26, 7.28, 7.30, Exercises 7.27, 7.29, 7.31, 7.32, 7.34, Remark 7.33; CTfS §3.4.4.8 (basic open sets, Example 3.4.4.9, Exercises 3.4.4.10–3.4.4.12), Example 4.2.3.1, Exercise 4.2.3.2, Example 4.2.3.3

Examples

• Metric spaces (Example 7.26): in ${\mathbb{R}}^2$ (or any Metric Space) the $ϵ$-ball is $B(p;ϵ)=\{p^{\prime }\mid d(p,p^{\prime })<ϵ\}$; $U$ is open iff every $p\in U$ has some $B(p;ϵ)\subseteq U$. On $\mathbb{R}$: $B(x,ϵ)=(x-ϵ,x+ϵ)$; $(0,2)$ and $(1,3)$ cover $(0,3)$; ${\bigcup }_{i\ge 1}(\frac{1}{i},1)=(0,1)$ is an infinite cover (7S Exercise 7.27).
• Coarse and discrete (Example 7.28): ${\mathrm{Op}}_{\mathrm{crse}}=\{\varnothing ,X\}$ has the fewest opens; ${\mathrm{Op}}_{\mathrm{fine}}=\mathcal{P}(X)$ the most — the discrete space, from which every function is continuous (7S Exercise 7.29).
• Sierpiński space (Example 7.30): $X=\{1,2\}$ with ${\mathrm{Op}}_1=\{\varnothing ,\{1\},X\}$ (or the isomorphic ${\mathrm{Op}}_2$); the two remaining topologies on $\{1,2\}$ are coarse and discrete. See Sierpinski Space.
• Subspace topology (7S Exercise 7.32): for $Y\subseteq X$, $A\subseteq Y$ is open iff $A=B\cap Y$ for some $B\in \mathrm{Op}$; the inclusion $Y\hookrightarrow X$ is then continuous.
• The Interval Domain $\mathbb{IR}$, the site of the Topos of Behavior Types.

Spaces in Category Theory for Scientists

CTfS introduces open sets through basic open sets — interiors of closed curves such as circles, ellipses or kidney beans — and unions of them: the half-plane $\{(x,y)\mid x>0\}$ is the union of the open unit squares with upper-left corner $(a,b)$, $a>0$ (CTfS Example 3.4.4.9). “On the earth one could define a basic open set to be the interior of any region one can draw a circle around.” A topology is a sub-order $\mathrm{Op}(X)\subseteq \mathcal{P}(X)$ closed under finite meets and arbitrary joins; continuity says preimage restricts to $\mathrm{Op}(Y)\to \mathrm{Op}(X)$, so “looking at points” is a functor $\mathbf{Top}\to \mathbf{Set}$ while “looking at open sets” is a functor ${\mathbf{Top}}^{\mathrm{op}}\to \mathbf{Preord}$ (CTfS Exercise 4.2.3.2). Assigning to each region the interval of recorded temperatures is a monotone map preserving joins but not meets (Join); a functor from the topological monoid $(\mathbb{R},0,+)$ to $\mathbf{Top}$ is a continuous dynamical system (CTfS Example 4.2.3.3); paths up to homotopy form the fundamental Groupoid.

The preorder of open sets

$(\mathrm{Op},\subseteq )$ is a Preorder (indeed a Partial Order), hence a Category: one morphism $U\to V$ iff $U\subseteq V$. A Presheaf on $\mathrm{Op}$ assigns sets of sections to opens with restriction maps; a Sheaf is a presheaf respecting covers. Moreover $(\mathrm{Op},\subseteq ,X,\cap )$ is a Quantale (Remark 7.33) and a Heyting Algebra: a $\mathrm{Op}$-enriched category has “size restrictions” $\mathcal{C}(a,b)\in \mathrm{Op}$ like bridges a truck must fit under (7S Exercise 7.34).

JuliaLeanHaskell

Docs: ThThinCategory (GATlab) · Theories & presentations

# a finite topology as a set of BitSets, and the axioms checked by brute force
X = 1:2
Op1 = Set([BitSet(), BitSet([1]), BitSet([1, 2])])        # Sierpiński
istopology(X, Op) = BitSet(X) in Op && BitSet() in Op &&
  all(union(U, V) in Op && intersect(U, V) in Op for U in Op, V in Op)
istopology(X, Op1)                                        # true
istopology(X, Set([BitSet(), BitSet([1]), BitSet([2])]))  # false: missing the whole set

Catlab version (run in a fresh Julia session — Catlab exports its own compose, id, FinFunction, …):

# in Catlab the poset Op is a thin category; e.g. Sierpiński as a FreePreorder presentation
using Catlab
@present Op1Cat(FreePreorder) begin (∅, U, X)::El; a::Leq(∅, U); b::Leq(U, X) end

import Mathlib
#check @TopologicalSpace              -- class: IsOpen, isOpen_univ, isOpen_inter, isOpen_sUnion
#check @Continuous                    -- ∀ U, IsOpen U → IsOpen (f ⁻¹' U)
#check @TopologicalSpace.Opens        -- the poset (frame) of open sets
#check @sierpinskiSpace               -- TopologicalSpace Prop
#check @instTopologicalSpaceSubtype   -- subspace topology
#check @Metric.isOpen_iff             -- ε-ball characterization

import qualified Data.Set as S
type Topology a = S.Set (S.Set a)
isTopology :: Ord a => S.Set a -> Topology a -> Bool
isTopology x op = S.member x op && S.member S.empty op
  && and [ S.member (S.union u v) op && S.member (S.intersection u v) op | u <- S.toList op, v <- S.toList op ]