Let be $f$ a perfect map from a space $X$ to a space $Y$ so that I am trying to prove or disprove if the inequality $$ \begin{equation}\tag{1}\label{1}w(Y)\le w(X)\end{equation} $$ holds, where $w$ is the weight. So, on this purpose, let be $\Lambda$ a collection with cardinality $\lambda$ and thus let we observe that the collection $\scr U(\Lambda)$ of finite union of $\Lambda$ has cardinality $\lambda$ because obviously $\Lambda$ is a subset of $\scr U(\Lambda)$ and because any finite union can be identified with a function from any $n\in\omega$ to $\lambda$ and viceversa so that $$ |\mathscr U(\Lambda)|=\biggl|\bigcup_{n\in\omega}\lambda^n\biggl|\le\sum_{n\in\omega}\lambda^n\le\sum_{n\in\omega}\lambda^{n+1}=\sum_{n\in\omega}\lambda=\lambda\cdot\aleph_0=\lambda $$ So let be $\mathcal B_X$ a base whose cardinality is $w(X)$ and thus let's we prove that the collection $$ \mathcal B_Y:=\Big\{Y\setminus f\big[X\setminus U\big]:U\in \mathscr U(\mathcal B_X)\Big\} $$ is a base for $Y$ whose cardinality is $w(X)$ so that let be $y$ and element of any open set $V$ of $Y$. So we observe that the inclusion $$ f^{-1}[y]\subseteq f^{-1}[V] $$ but continuity implies that $f^{-1}[V]$ is open so that for any $x\in f^{-1}[y]$ there exists $B_x\in\mathcal B_X$ such that $$ x\in B_x\subseteq f^{-1}[V] $$ so that by compactness of $f^{-1}[y]$ there exists $U\in\mathscr U(\mathcal B_X)$ such that $$ f^{-1}[y]\subseteq U\subseteq f^{-1}[V] $$ which implies that $$ y\in Y\setminus f\big[X\setminus U\big]\subseteq f[U]\subseteq V $$ so that we conclude that $\mathcal B_Y$ is a base. After all, $\mathcal B_Y$ is indexed on $\mathscr U(\mathcal B_X)$ whose cardinality is $w(X)$. So we conclude that $$ w(Y)\le w(X) $$
So first of all I ask if $\eqref{1}$ holds and so if it holds I ask if I prove it well. Moreover I ask if $f^{-1}[y]$ is Lindelöf then is it possible to relate $w(X)$ with $w(Y)$? indeed in this case $\mathscr U(\Lambda)$ became the subset of countable union of $\Lambda$ but I only know that $$ \lambda\le\lambda^{\aleph_0} $$ and the equality cannot be false so that I am not able to say something about $w(Y)$. Finally is it possible for $\eqref{1}$ be always a equality? So could someone help me, please?