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

In summary, The equation states that if E is an open set then ∂E is also an open set, but if E is closed then ∂E is also a closed set.
  • #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?
 
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  • #2
There is no need to make a separate itex-tag for each math symbol. Things like this

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

work fine.
 
  • #3
tsuwal said:

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?

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

1. What is a metric space?

A metric space is a mathematical structure that consists of a set of elements and a function that measures the distance between any two elements in the set. The function, called a metric, must follow certain properties such as non-negativity, symmetry, and the triangle inequality.

2. What is the purpose of Exercise 1.14 from Introduction to Topology?

The purpose of Exercise 1.14 is to provide practice in understanding and working with the definition of a metric space, as well as to develop skills in verifying that a given set and metric satisfy the axioms of a metric space.

3. How do you show that a set and metric satisfy the axioms of a metric space?

To show that a set and metric satisfy the axioms of a metric space, you must verify that the metric function satisfies the properties of non-negativity, symmetry, and the triangle inequality for any two elements in the set. This can be done by plugging in the elements into the metric function and showing that the resulting values satisfy the axioms.

4. Can you give an example of a metric space?

Yes, an example of a metric space is the set of real numbers with the metric function of absolute value. In this metric space, the distance between any two real numbers is given by the absolute value of their difference. This satisfies the axioms of a metric space.

5. What is the difference between a metric space and a topological space?

A metric space and a topological space are both mathematical structures used to describe the properties of sets. However, a metric space uses a specific function, the metric, to measure the distance between elements, while a topological space uses a set of open sets to define the structure of the space. Additionally, all metric spaces are also topological spaces, but the converse is not always true.

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