
4. CONTINUITY 11
4.1. Directed topological spaces. Directed topological spaces are defined
at
http://ncatlab.org/nlab/show/directed+topological+space
Definition 2086. A directed topological space (or d-space for short) is a pair
(X, d) of a topological space X and a set d ⊆ C([0; 1], X) (called directed paths or
d-paths) of paths in X such that
1
◦
. (constant paths) every constant map [0; 1] → X is directed;
2
◦
. (reparameterization) d is closed under composition with increasing con-
tinuous maps [0; 1] → [0; 1];
3
◦
. (concatenation) d is closed under path-concatenation: if the d-paths a,
b are consecutive in X (a(1) = b(0)), then their ordinary concatenation
a + b is also a d-path
(a + b)(t) = a(2t), if 0 ≤ t ≤
1
2
,
(a + b)(t) = b(2t − 1), if
1
2
≤ t ≤ 1.
I propose a new way to construct a directed topological space. My way is more
geometric/topological as it does not involve dealing with particular paths.
Definition 2087. Let T be the complete endofuncoid corresponding to a topo-
logical space and ν v T be its “subfuncoid”. The d-space (dir)(T, ν) induced by
the pair (T, ν) consists of T and paths f ∈ C([0; 1], T ) ∩ C(|[0; 1]|
≥
, ν) such that
f(0) = f(1).
Proposition 2088. It is really a d-space.
Proof. Every d-path is continuous.
Constant path are d-paths because ν is reflexive.
Every reparameterization is a d-path because they are C(|[0; 1]|
≥
, ν) and we
can apply the theorem about composition of continuous functions.
Every concatenation is a d-path. Denote f
0
= λt ∈ [0;
1
2
] : a(2t) and f
1
= λt ∈
[
1
2
; 1] : b(2t − 1). Obviously f
0
, f
1
∈ C([0; 1], µ) ∩ C(|[0; 1]|
≥
, ν). Then we conclude
that a + b = f
1
t f
1
is in f
0
, f
1
∈ C([0; 1], µ) ∩ C(|[0; 1]|
≥
, ν) using the fact that the
operation ◦ is distributive over t.
Below we show that not every d-space is induced by a pair of an endofuncoid
and its subfuncoid. But are d-spaces not represented this way good anything except
counterexamples?
Let now we have a d-space (X, d). Define funcoid ν corresponding to the d-
space by the formula ν =
d
a∈d
(a ◦ |R|
≥
◦ a
−1
).
Example 2089. The two directed topological spaces, constructed from a fixed
topological space and two different reflexive funcoids, are the same.
Proof. Consider the indiscrete topology T on R and the funcoids 1
FCD(R,R)
and 1
FCD(R,R)
t({0}×
FCD
∆
≥
). The only d-paths in both these settings are constant
functions.
Example 2090. A d-space is not determined by the induced funcoid.
Proof. The following a d-space induces the same funcoid as the d-space of all
paths on the plane.
Consider a plane R
2
with the usual topology. Let d-paths be paths lying inside
a polygonal chain (in the plane).