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location and lies in the second ring of regions around the core. The distance criterion is incorporated
in equation (4) indicating that the distance between the core and adjacent regions, θCA is less than the
distance between the core and the periphery regions, θCP. This also implies that the distance between a
periphery and an adjacent region θAP is less than the distance between the core and periphery regions,
such that θCP > θAP.
γ(updi ) ∈ P θ updi ∈ U ∩ φ(updi ) ∩ γ(updi )ΘΘcp > Θca ≥ Θap ) ∀i (4)
The extension theorem holds only if the following conditions are met. If φ(updi) →C ∪ (A ∪ P) = U,
γ(updi) →A ∪ (c ∪ P) = U, and γ(updi)(θcP > θcA ≥ θAP)→P ∪ (c ∪ A) = U, then:
∃ c∪A∪P(θcP >θcA ≥θAP)=U ∀i (5)
cP cA AP
and
∃ C ∩ A ∩ P(Θcp > Θca ≥ Θap ) = 0 ∀i (6)
cP cA AP
The regions are disjoint because of the urban population density - and distance criteria assigned to
each subset of regions. The regions are individual non-overlapping units bordering on each other in the
order as given by equation (6). The universal set of regions can be rewritten as follows:
U = ∪J R ∀i (7)
j=1 j
Then one may write,
J
U = ∪ R = c ∩ A ∩ P(ΘcP > ΘcA ≥ ΘAP ) = 0 ∀i (8)
1 j cP cA AP