�
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Fixed-Point Theory for Generalized Metric Spaces
4.3.8 Definition Let (x δ ) δ<η be a (possibly transfinite) sequence of elements
of a gum (X, �, Γ). Then (x δ ) is said to be pseudo-convergent if, for all α < β <
γ < η, we have �(x β , x γ ) < �(x α , x β ). The transfinite sequence (π δ ) δ+1<η with
π δ = �(x δ , x δ+1 ) is then strictly monotonic decreasing. If η is a limit ordinal,
then any x ∈ X with �(x, x δ ) ≤ π δ for all δ < η is called a pseudo-limit of the
transfinite sequence (x δ ) δ<η .
The space (X, �, Γ) is called trans-complete if every pseudo-convergent
transfinite sequence (x δ ) δ<η , where η is a limit ordinal, has a pseudo-limit in
X.
4.3.9 Proposition Suppose that x is a pseudo-limit of (x δ ) δ<η , where η is a
limit ordinal. Then the set of all pseudo-limits of (x δ ) is given by Lim(x δ ) =
{z ∈ X | �(x, z) < π δ for all δ < η}.
Proof: Let z ∈ Lim(x δ ). Since �(z, x) < π δ and �(x, x δ ) ≤ π δ , we obtain
�(z, x δ ) ≤ π δ for all δ, and hence z is a pseudo-limit. Conversely, let z be
a pseudo-limit of (x δ ). Since �(x, x δ+1 ), �(z, x δ+1 ) ≤ π δ+1 for all δ < η, we
obtain �(x, z) ≤ π δ+1 < π δ for all δ < η, as required.
•
4.3.10 Proposition A generalized ultrametric space is spherically complete
if and only if it is trans-complete.
Proof: Let X be trans-complete, and let B be a decreasing chain of balls in X.
Without loss of generality, assume that B does not have a minimal element and
is, in fact, strictly decreasing. Then we can select a coinitial subchain (B δ ) δ<η
of B, where η is a limit ordinal, so that (B δ ) δ<η is a transfinite sequence of
balls. Since this transfinite sequence is strictly decreasing, we know that for
every δ there exists x δ ∈ B δ \ B δ+1 , and the transfinite sequence (x δ ) δ<η is
pseudo-convergent; hence, it has a pseudo-limit x. Since �(x, x δ ) ≤
� �(x δ , x δ+1 )
and x δ , x δ+1 ∈ B δ , we obtain x ∈ B δ for all δ, and therefore, x ∈ B.
Conversely, let X be spherically complete, and let (x δ ) be pseudoconvergent. Let π δ = �(x δ , x δ+1 ), and let B δ = B π δ (x δ ). For α < β, we
have that x β ∈ B α ∩ B β , and therefore (B δ ) is a decreasing chain of balls by
Lemma 4.3.5. By spherical completeness, there is some x ∈ B δ , and it is
immediate that x is a pseudo-limit of (x δ ).
•
We close this section by considering briefly how pseudo-convergent sequences may be generated when the set Γ is linearly ordered. Thus, in what
follows, let (X, �, Γ) be a generalized ultrametric space in which Γ is a linearly
ordered set.
4.3.11 Lemma Let x, y, z ∈ X with �(x, y) < �(y, z). Then �(x, z) = �(y, z).
Proof: We have �(x, z) ≤ max{�(x, y), �(y, z)} ≤ �(y, z) on using the strong
triangle inequality. Now assume �(y, z) ≤ �(x, z). Then, because Γ is linearly
101
Fixed-Point Theory for Generalized Metric Spaces
4.3.8 Definition Let (x δ ) δ<η be a (possibly transfinite) sequence of elements
of a gum (X, �, Γ). Then (x δ ) is said to be pseudo-convergent if, for all α < β <
γ < η, we have �(x β , x γ ) < �(x α , x β ). The transfinite sequence (π δ ) δ+1<η with
π δ = �(x δ , x δ+1 ) is then strictly monotonic decreasing. If η is a limit ordinal,
then any x ∈ X with �(x, x δ ) ≤ π δ for all δ < η is called a pseudo-limit of the
transfinite sequence (x δ ) δ<η .
The space (X, �, Γ) is called trans-complete if every pseudo-convergent
transfinite sequence (x δ ) δ<η , where η is a limit ordinal, has a pseudo-limit in
X.
4.3.9 Proposition Suppose that x is a pseudo-limit of (x δ ) δ<η , where η is a
limit ordinal. Then the set of all pseudo-limits of (x δ ) is given by Lim(x δ ) =
{z ∈ X | �(x, z) < π δ for all δ < η}.
Proof: Let z ∈ Lim(x δ ). Since �(z, x) < π δ and �(x, x δ ) ≤ π δ , we obtain
�(z, x δ ) ≤ π δ for all δ, and hence z is a pseudo-limit. Conversely, let z be
a pseudo-limit of (x δ ). Since �(x, x δ+1 ), �(z, x δ+1 ) ≤ π δ+1 for all δ < η, we
obtain �(x, z) ≤ π δ+1 < π δ for all δ < η, as required.
•
4.3.10 Proposition A generalized ultrametric space is spherically complete
if and only if it is trans-complete.
Proof: Let X be trans-complete, and let B be a decreasing chain of balls in X.
Without loss of generality, assume that B does not have a minimal element and
is, in fact, strictly decreasing. Then we can select a coinitial subchain (B δ ) δ<η
of B, where η is a limit ordinal, so that (B δ ) δ<η is a transfinite sequence of
balls. Since this transfinite sequence is strictly decreasing, we know that for
every δ there exists x δ ∈ B δ \ B δ+1 , and the transfinite sequence (x δ ) δ<η is
pseudo-convergent; hence, it has a pseudo-limit x. Since �(x, x δ ) ≤
� �(x δ , x δ+1 )
and x δ , x δ+1 ∈ B δ , we obtain x ∈ B δ for all δ, and therefore, x ∈ B.
Conversely, let X be spherically complete, and let (x δ ) be pseudoconvergent. Let π δ = �(x δ , x δ+1 ), and let B δ = B π δ (x δ ). For α < β, we
have that x β ∈ B α ∩ B β , and therefore (B δ ) is a decreasing chain of balls by
Lemma 4.3.5. By spherical completeness, there is some x ∈ B δ , and it is
immediate that x is a pseudo-limit of (x δ ).
•
We close this section by considering briefly how pseudo-convergent sequences may be generated when the set Γ is linearly ordered. Thus, in what
follows, let (X, �, Γ) be a generalized ultrametric space in which Γ is a linearly
ordered set.
4.3.11 Lemma Let x, y, z ∈ X with �(x, y) < �(y, z). Then �(x, z) = �(y, z).
Proof: We have �(x, z) ≤ max{�(x, y), �(y, z)} ≤ �(y, z) on using the strong
triangle inequality. Now assume �(y, z) ≤ �(x, z). Then, because Γ is linearly