Metric Spaces: Exercise 1.14 from Introduction to Topology (Dover)

[tsuwal]

Homework Statement

X is a metric and E is a subspace of X (E\subsetX)
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=\overline{E}\cap(\overline{X\E})
(ignore the red color, i can't get it out)

Show that E is open if and only if E\cap∂E is empty. Show that E is closed if and only if ∂E \subseteq E

Homework Equations

∂E=\overline{E}\cap(\overline{X\E})
(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 (\overline{X\E})=E , so,
∂E = \overline{E}\cap(\overline{X\E}) = \overline{E}\capE= empty set

Can anyone explain this to me?

 

[micromass]

There is no need to make a separate itex-tag for each math symbol. Things like this

[itex]E\cap \overline{X\setminus E}[/itex]

work fine.

 

[Dick]

Homework Statement

X is a metric and E is a subspace of X (E\subsetX)
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=\overline{E}\cap(\overline{X\E})
(ignore the red color, i can't get it out)

Show that E is open if and only if E\cap∂E is empty. Show that E is closed if and only if ∂E \subseteq E

Homework Equations

∂E=\overline{E}\cap(\overline{X\E})
(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 (\overline{X\E})=E , so,
∂E = \overline{E}\cap(\overline{X\E}) = \overline{E}\capE= empty set

Can anyone explain this to me?

You are probably thinking the bar means complement. It doesn't. It means closure. Look up the definition and using it explain why you think $\overline{X\backslash E}=E$. For the red tex problem use \backslash instead of \.

 

[tsuwal]

Thanks a lot guys! Without you I couldn't do my self-study!