Suppose X is a regular space.
Hence, the fixed point set of a resolution is always a retract of its convergence set, and when X is a regular space, every virtually stable scheme on X having a continuous subsequence is always a resolution.
Let X be a regular topological space and (Gamma :Xrightrightarrows X) be a u.s.c.s.c
Let ((mathcal X,d,preceq )) be a regular ordered metric space and let (mathcal F ) be a self-map on (mathcal X ) satisfying for all (x,yin overline{mathcal{O (x_0;mathcal F )}}) such that (x) and (y) are comparable, begin{aligned} T(d(mathcal F x,mathcal F y),d x,y),d x,mathcal F x),d y,mathcal F y),d x,mathcal F y),d y,mathcal F x))le 0 end{aligned}where (T in mathfrak T ^{prime }).
Corollary 5 Let ( X, d, ≼ ) be a regular ordered metric space and let T be a self-map on X satisfying for all x, y ∈ O ( x 0 ; T ) - such that x and y are comparable, F ( d ( T x, T y ) ) ≤ φ ( F ( M 3 [ T ] ( x, y ) ) ), where F ∈ F, φ ∈ Φ and M 3 [ T ] ( x, y ) = max { d ( x, y ), d ( T x, x ), d ( T y, y ), 1 2 [ d ( x, T y ) + d ( T x, y ) ] }.
We will denote by Φ the set of functions φ : [0 + ∞) → [0, +∞), such that φ is right continuous, φ(0) = 0 and φ(t) < t for every t > 0. The first result of this section is the following. Theorem 4 Let ( X, d, ≼ ) be a regular ordered metric space and let T, S and R be self-maps on X satisfying F ( d ( T x, S y ) ) ≤ φ ( F ( M [ T, S, R ] ( x, y ) ) ) (4.1).
Corollary 4 Let ( X, d, ≼ ) be a regular ordered metric space and let T and R be self-maps on X satisfying F ( d ( T x, T y ) ) ≤ φ ( F ( M 2 [ T, R ] ( x, y ) ) ), for all x, y ∈ O x 0 ; T, T, R - (for some x0) such that R x and R y are comparable, where F ∈ F, φ ∈ Φ and M 2 [ T, R ] ( x, y ) = max { d ( R x, R y ), d ( T x, R x ), d ( T y, R y ), 1 2 [ d ( R x, T y ) + d ( T x, R y ) ] }.
Let ((E,preceq, Vert cdot Vert )) be a regular partially ordered complete normed linear space.
Let ((X,leq,|cdot|)) be a regular partially ordered complete normed linear space such that the order relation ≤ and the norm (|cdot| ) in X are compatible.
The official did not say where in North Korea the preparations were taking place, but said people on the ground appeared to be readying for "a regular space launch".
