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

1. Feb 16, 2013

### tsuwal

1. The problem statement, all variables and given/known data
X is a metric and E is a subspace of X (E$\subset$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=$\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

2. Relevant equations

∂E=$\overline{E}$$\cap$($\overline{X\E}$)
(ignore the red color, i can't get it out)

3. 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}$$\cap$($\overline{X\E}$) = $\overline{E}$$\cap$E= empty set

Can anyone explain this to me?

2. Feb 16, 2013

### micromass

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

Code (Text):

$E\cap \overline{X\setminus E}$

work fine.

3. Feb 16, 2013

### Dick

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 \.

4. Feb 16, 2013

### tsuwal

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