Add conditions on Tychonoff space $X$ that guarantee that $X$ is perfectly normal. There exist Tychonoff spaces $X$ that are not perfectly normal. For example Uncountable Fort Space is a Tychonoff normal space which is not perfectly normal. Can we add conditions (maybe in cardinal invariants) on a Tychonoff space $X$ such that $X$ is perfectly normal?
 A: In fact, cardinal invariants are used already in the definition of a perfectly normal space, that is a space which is perfect and normal. A space $X$ is perfect if  $\psi(F,X)\le\omega$ for each closed subset $F$ of $X$. 
Each regular semistratifiable space is perfect (in particular, each regular $\sigma$-space, each regular stratifiable space, each regular $\aleph$-space, each regular $M_1$-space, each regular Lasnev space) (see the Chapter “Generalized metric spaces” in “Handbook of Set-Theoretic Topology” for the definitions of these spaces).
The class of perfectly normal spaces contains: all discrete spaces ($\chi(X)=1$); all countable normal spaces ($|X|\le\omega$); all metrizable spaces (here you can use one of many metrization criteria, for instance, Bing-Nagata-Smirnov Theorem: a regular space $X$ is metrizable iff $X$ has a $\sigma$-locally finite (Bing, $\sigma$-discrete) base).
Also you can look at pages indexed with “perfectly normal space” in Engelking’s “General topology” for more complex conditions for the perfect normality of the space. 
A: The following well-known examples which are perfectly normal may be helpful for the question:

1 Metric space.
2 Topologist’s Sine Curve
3 Right Half-Open Interval Topology
4 Discrete space

