  We start the systematic study of Fr\'{e}chet spaces which are $\aleph$-spaces
in the weak topology. A topological space $X$ is an $\aleph_0$-space or an
$\aleph$-space if $X$ has a countable $k$-network or a $\sigma$-locally finite
$k$-network, respectively. We are motivated by the following result of Corson
(1966): If the space $C_{c}(X)$ of continuous real-valued functions on a
Tychonoff space $X$ endowed with the compact-open topology is a Banach space,
then $C_{c}(X)$ endowed with the weak topology is an $\aleph_0$-space if and
only if $X$ is countable. We extend Corson's result as follows: If the space
$E:=C_{c}(X)$ is a Fr\'echet lcs, then $E$ endowed with its weak topology
$\sigma(E,E')$ is an $\aleph$-space if and only if $(E,\sigma(E,E'))$ is an
$\aleph_0$-space if and only if $X$ is countable. We obtain a necessary and
some sufficient conditions on a Fr\'echet lcs to be an $\aleph$-space in the
weak topology. We prove that a reflexive Fr\'echet lcs $E$ in the weak topology
$\sigma(E,E')$ is an $\aleph$-space if and only if $(E,\sigma(E,E'))$ is an
$\aleph_0$-space if and only if $E$ is separable. We show however that the
nonseparable Banach space $\ell_{1}(\mathbb{R})$ with the weak topology is an
$\aleph$-space.
