Factual error: Titan is depicted as a Moon-like landscape with no atmosphere. In actual fact, Titan has a nitrogen/argon atmosphere denser than Earth's, and a reddish-orange sky. This was detected almost thirty years before "The Invisible Enemy" was made, so somebody slipped up in doing their research.
Underworld - S15-E5
Factual error: While it's true the centre of a planet would have no net gravity, the weightless condition would be everywhere, not concentrated in a convenient shaft.
Other mistake: The Brigadier says UNIT HQ is 'a Top Secret establishment'. It is, in fact, so secret that it has a large sign outside informing the world not only of its function, but also the name of the commanding officer.
The Doctor: You can't rule the world in hiding. You've got to come out on the balcony sometimes and wave a tentacle, if you pardon the expression.
Planet of the Spiders - S11-E5
Trivia: The first thing the Third Doctor does on-screen is collapse out of the TARDIS, which is also the last thing he does in that incarnation.
Question: If the Doctor gets a TARDIS, does that mean other Timelords get a TARDIS as well?
Answer: TARDISes are generally available for properly authorised use on Gallifrey; they're not usually assigned to a particular Timelord on a long-term basis. The Doctor stole his when he left his homeworld.
Tailkinker ★