# Thread: More basic topology (couple questions)

1. ## More basic topology (couple questions)

1) I have to prove: $\displaystyle (A \cup B) - (A \cap B) = (A-B) \cup (B-A)$

I know some people get annoyed by the "-" notation, but that is what my prof uses, so...

Here is my work:

$\displaystyle (A \cup B) - (A \cap B) = (A \cup B) \cap (A \cap B)^c$

$\displaystyle =(A \cup B) \cap (A^c \cup B^c)$

$\displaystyle =(A \cup B) \cup (B \cap A^c)$

$\displaystyle =(A-B) \cup (B-A)$

2) Let $\displaystyle X = \{ a,b,c,d \}$

and let $\displaystyle S = \{ \O , \{ a,b \} , \{ b,c \} , \{ c,d \} , \{ a,d \} , \{ a,b,c,d \} \}$

I have to find the smallest topology on $\displaystyle X$ that contains $\displaystyle S$.

So, using the axioms for a topological space:

$\displaystyle \{ \{ S, \{ a \} , \{ b \} , \{ c \} , \{ d \} , \{ b,d \} , \{ a,c \} , \{ a,b,c \} , \{ a,b,d \} , \{ b,c,d \} , \{ a,c,d \} \}$

Correct?

2. You missed an important equality.
$\displaystyle (A \cup B) \cap (A^c \cup B^c)=(A\cap B^c)\cup(B\cap A^c)$.

3. Oh right, so it should have been:

$\displaystyle (A \cup B) - (A \cap B) = (A \cup B) \cap (A \cap B)^c$

$\displaystyle =(A \cup B) \cap (A^c \cup B^c)$

$\displaystyle =(A \cap B^c) \cup (B \cap A^c)$

$\displaystyle =(A-B) \cup (B-A)$