>>11068544Give some examples, and make sure you're not confusing proof of negation with proof by contradiction.
Negation: Prove ~P by showing P leads to contradiction (this is actually the definition of ~P in intuitionistic logic, ie the logic of constructivist mathematics)
Contradiction: Prove P by showing ~P leads to contradiction (ie, prove ~~P directly, and then use double negation to conclude P, that being valid in classical logic, but not IL)
So in topology, connected means not separated, thus the way to prove a space is connected is to show that a separation leads to a contradiction. But this would just be a direct proof(!) that a space is not separated. Don't be confused by the fact a contradiction is involved. That's just what it means for ~P to hold (P implies a contradiction).
