aa r X i v : . [ m a t h . G M ] N ov Directed and irreversible path in Euclidean spaces
Khashayar RahimiNovember 12, 2020
Abstract
The aim of this very short note is to relate the directed paths in −→ R n to the irreversible paths in ir R n .We first show that there is a directed path from x to y in −→ R n iff there exists an irreversible path withsame initial and terminal points in ir R n . Also, we prove that every directed path in −→ R n is an irreversiblepath in ir R n . Now we briefly present some preliminaries including definitions and notations.
Definition 1. ([1]) A directed topological space, or a d-space X = ( X, P X ) is a topological spaceequipped with a set
P X of continuous maps γ : I → X (where I = [0 ,
1] equipped with subspace topologyof standard topology on R ), called directed paths or d-paths, satisfying these axioms:1. every constant map I → X is directed2. P X is closed under composition with continuous non-decreasing maps from I to I P X is closed under concatenation
Notation ([2]) ir R is the real line equipped with the left order topology, and ir I is [0 ,
1] equipped withsubspace topology of ir R . Definition 2.
Let X be a topological space. A function γ : ir I → X is called an ir-path in X , if it iscontinuous on ir I. Also, γ (0) is the initial point and γ (1) is the terminal point of the ir-path γ . Definition 3. ([3]) −→ Γ X = (cid:26) ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) ∃ γ ∈ P X, γ (0) = x, γ (1) = y (cid:27) Definition 4. ir Γ X = (cid:26) ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) ∃ γ ∈ X ir I , γ (0) = x, γ (1) = y (cid:27) Proposition 1. −→ Γ −→ R n = ir Γ ir R n . Proof.
From [4] we know that 1 → Γ −→ R n = (cid:26) ( x . . . x n , y . . . y n ) (cid:12)(cid:12)(cid:12)(cid:12) x i ≤ y i ∀ i ∈ { , . . . , n } (cid:27) Also, we know from [2] that there exists an ir-path from x = ( x . . . x n ) ∈ ir R n to y = ( y . . . y n ) ∈ ir R n iff y ∈ { x } . Therefore ( x, y ) ∈ ir Γ ir R n iff ( y . . . y n ) ∈ Q ni =1 [ x i , ∞ ). Clearly, for all i ∈ { , . . . , n } we have x i ≤ y i ,which proves the statement. (cid:3) Theorem 1.
Every d-path in −→ R n is an ir-path in ir R n . Proof.
We know that d-paths in −→ R n are non-decreasing paths in R n . Thus, it suffices to show that everynon-decreasing path in R n is an ir-path in ir R n .Suppose that γ : I → R n is a non-decreasing path from x = ( x . . . x n ) to y = ( y . . . y n ) and Q ni =1 ( −∞ , m i ) is a base element of ir R n . Now, if γ − (cid:18) Q ni =1 ( −∞ , m i ) (cid:19) is an open subset of ir I, then γ is anir-path in ir R n . There are two cases for m i . If for some 1 ≤ i ≤ n , m i ≤ x i , then γ − (cid:18) Q ni =1 ( −∞ , m i ) (cid:19) = ∅ is open in ir I. If for all 1 ≤ i ≤ n , m i ≥ x i , then 0 ∈ γ − (cid:18) Q ni =1 ( −∞ , m i ) (cid:19) . Also we know that γ − (cid:18) Q ni =1 ( −∞ , m i ) (cid:19) is open in I . Hence, γ − (cid:18) Q ni =1 ( −∞ , m i ) (cid:19) = [0 , n ) is open in ir I.The case γ − (cid:18) Q ni =1 ( −∞ , m i ) (cid:19) = [0 , n ) ∪ U = [0 , p ) does not happen. Consider an element u ∈ U . Since γ is non-decreasing, γ ( n ) ≤ γ ( u ) < ( m . . . m n ), then n ∈ γ − (cid:18) Q ni =1 ( −∞ , m i ) (cid:19) , a contradiction. (cid:3) Example.
This example demonstrates that there exist ir-paths in ir R n which are not d-paths in −→ R n .Let γ : ir I → ir R n be an ir-path from x = ( x . . . x n ) to y = ( y . . . y n ) defined by γ ( t ) = (cid:26) x ≤ t < y t = 1Since γ − (cid:18) Q ni =1 ( x i , y i + ǫ ) (cid:19) = { } is not open in I , γ : I → R n is not continuous, and hence not a path.Therefore, γ is an ir-path that is not a d-path. References [1] Marco Grandis,
Directed Algebraic Topology, Models of non-reversible worlds , Cambridge Uni-versity Press, 2009.[2] Khashayar Rahimi,
Irreversible homotopy and a notion of irreversible Lusternik–Schnirelmanncategory , arXiv preprint arXiv:2010.11217, 2020.[3] E. Goubault,
On directed homotopy equivalences and a notion of directed topological complexity ,arXiv preprint arXiv:1709.05702, 2017.[4] A.Borat and M.Gran,