Isn't an improper subset of edges of a cyclic graph, cyclic and thus not a minimum spanning tree?
It looks like you misread or misunderstand the standard set notation &quot;$E'\subseteq E$&quot;.
That notation just means $E'$ is a subset of $E$, i.e. every element of $E'$ is also an element of $E$.

We do not speak of &quot;an improper subset&quot;, most of the time if not ever.
Why?
Suppose $A$ and $B$ are two sets. $A$ is a subset of $B$ iff every element of $A$ is also an element of $B$. If, furthermore, $A$ is not $B$, then we say $A$ is a proper (or strict) subset of $B$.
If we want to mention &quot;an improper subset&quot; of $B$, what would it mean?

It should mean $B$ itself. That is how I would understand that phrase upon first reading. Then, why do we want to say &quot;an improper subset of $B$&quot;, a pretty long phrase, when we can just say &quot;$B$&quot;, which is concise? The phrase &quot;an improper subset&quot; seems implying there are more-than-one such subsets for a given set, which is simply wrong.

It could just means &quot;subset&quot; when it is used in distinction to &quot;proper subset&quot; (which sounds unnatural to me, but maybe that is only me). Again, why not just use &quot;subset&quot;, which is concise?