Quotient spaces and multiplicity of a base

The basic results of the note are the following two theorems.
Theorem 1.1. Let $f\colon X\to Y$ be a biquotient $\tau$-mapping and let the space $X$ have a base whose multiplicity does not surpass $\tau$. Then the space $Y$ also has a base whose multiplicity does not surpass $\tau$. \smallskip
Theorem 2.1. Let $f\colon X\to Y$ be a quotient $s$-mapping of a space $X$ with a pointwise-countable base on a $T_2$-space $Y$ of pointwise-countable type. Then the mapping $f$ is biquotient.
References: 9 titles.