TITLE: Topological properties such that if the subspace has P then the whole space also has P
QUESTION [5 upvotes]: Is there any topological Property $P$ if it is satisfied on a subspace topology $N$ of a topological space $X$, then it must be satisfied on a topological space $X$. Thanks.

REPLY [15 votes]: Not first-countable. Not second-countable. Not Hausdorff. Not regular. Not completely regular. Not T0. Not T1. Not metrizable.
In general, if $\mathfrak{P}$ is a topological property inherited by subspaces, then the topological property not-$\mathfrak{P}$ is of the kind sought.

REPLY [10 votes]: Recall that a topological property is called hereditary if the following holds: If $X$ has the property P and $N$ is subspace of $X$, then $N$ also has the property P.
If you take any hereditary property P, the negation of it $\neg P$ will fulfill the conditions you ask in your post.
In fact, the property $P$ is hereditary if and only if $\neg P$ has fulfills the requirements from your post.

In some contexts, results of this type are might be called reflection theorems. However, as the two formulations are equivalent, they can be formulated as in your post, or dually. Quite often some additional condition is given on the subspace.
The following quote is from Encyclopedia of General Topology; Edited by: Klaas Pieter Hart, Jun-iti Nagata and Jerry E. Vaughan, Elsevier, 2003, p.16.

Generally speaking, a reflection theorem is a theorem of the form "if a set (or a space) $X$ has a property $P$, then there is a subset (or a subspace) $Y$ of a small size (in some sense) satisfying $P$.