Subspaces of a Completely Normal Space Completely normality is equivalent to the requirement that every subspace is normal. If we think of completely normality as a restriction of normality to obtain the normality of subspaces, then there should be a stricter condition $P$ than complete normality satisfying the following property: _$P(X)$ if and only if all subspaces in $X$ are completely normal._ Now if we keep this process going, should we ultimately find a property $P_n$ such that every subspaces in a space $X$ satisfying $P_n$ also satisfy $P_n$?

A subspace $Y$ of a completely normal space $X$ is again completely normal: if $Z$ is a subspace of $Y$, then $Z$ is also a subspace of $X$ and so $Z$ is normal.

Another name for this property is "hereditarily normal". In general we can define, when we have a topological property $P$, the (probably) stronger property "hereditarily P": all subspaces of $X$ have property $P$.

For $P$ = "normal" this indeed is a stronger property than $P$. Hereditarily Lindelöf spaces are also a standard property, as are hereditarily separable spaces. In all these cases there are spaces with property $P$ that are not hereditarily $P$.

The process (if it can be called that) stops here though: a space that is hereditarily $P$ is also "hereditarily hereditarily $P$", using the same remark that a subspace of a subspace is still a subspace of the original space (transitivity of subspace topologies).