# Preliminary Results

• Fausto Di Biase
Part of the Progress in Mathematics book series (PM, volume 147)

## Abstract

Let (W, ρ) be a quasi-metric space. A set D is a space of approach to (W, ρ), for the approach function $$\tilde \rho$$: D x W →[0, ∞), if
1. 1.
for each wW there is a sequence $${\{ {z_n}\} _n}$$ in D such that
$$\mathop {\lim }\limits_{n \to \infty } \tilde \rho \left( {{z_n},w} \right) = 0$$

1. 2.
whenever
$$\mathop {\lim }\limits_{n \to \infty } \rho \left( {{w_n},{u_n}} \right) = 0{\text{ }}and{\text{ }}\mathop {\lim }\limits_{n \to \infty } \tilde \rho \left( {{z_n},{w_n}} \right) = 0$$
then
$$\mathop {\lim }\limits_{n \to \infty } \tilde \rho \left( {{z_n},{u_n}} \right) = 0$$
;

1. 3.

whenever

$$\mathop {\lim }\limits_{n \to \infty } \tilde \rho \left( {{z_n},{w_n}} \right) = \mathop {\lim }\limits_{n \to \infty } \tilde \rho \left( {{z_n},{u_n}} \right) = 0$$
then
$$\mathop {\lim }\limits_{n \to \infty } \rho \left( {{w_n},{v_n}} \right) = 0$$
, where {Zn|n is a sequence in D and {wn|n, {un|n are sequences in W.