Why is Baire cat. theorem equivalent as $\cap U_i$ is dense as $\cup U_i$ has no int points?

Question

Why is Baire cat. theorem equivalent as $$\bigcap U_i$$ of open, dense $U_i$ is dense

as $$\bigcup U_i$$ of closed, nowhere dense $U_i$s has no int points?

Answer 1

Consider complements ($A^c$ denotes the complement of $A$):

• $U$ is open iff $U^c$ is closed
• $U$ is dense iff $U^c$ has empty interior
• $\bigcap U_i$ = $(\bigcup U_i^c) ^c$

(and note that closed sets are nowhere dense iff they have empty interior)