- #1
tsuwal
- 105
- 0
Homework Statement
X is a metric and E is a subspace of X (E[itex]\subset[/itex]X)
The boundary ∂E of a set E is defined to be the set of points adherent to both E and the complement of E,
∂E=[itex]\overline{E}[/itex][itex]\cap[/itex]([itex]\overline{X\E}[/itex])
(ignore the red color, i can't get it out)
Show that E is open if and only if E[itex]\cap[/itex]∂E is empty. Show that E is closed if and only if ∂E [itex]\subseteq[/itex] E
Homework Equations
∂E=[itex]\overline{E}[/itex][itex]\cap[/itex]([itex]\overline{X\E}[/itex])
(ignore the red color, i can't get it out)
The Attempt at a Solution
To begin with I don't understand the equation because it seems to me that ([itex]\overline{X\E}[/itex])=E , so,
∂E = [itex]\overline{E}[/itex][itex]\cap[/itex]([itex]\overline{X\E}[/itex]) = [itex]\overline{E}[/itex][itex]\cap[/itex]E= empty set
Can anyone explain this to me?