5.2 Open And Closed Sets
Eventually, one must consider functions which are defined on subsets of Fn and their
properties. The next definition will end up being quite important. It describe a type of
subset of Fn with the property that if x is in this set, then so is y whenever y is
close enough to x. In all of this, for x a vector, |x| is given by (x,x)1∕2 where
this denotes the square root of the inner product of the vector with itself as
described earlier. Then the distance between the vectors x and y is defined as
|x − y|.
Definition 5.2.1 Let U ⊆ Fn. U is an open set if whenever x ∈ U, there exists
r > 0 such that B
⊆ U. More generally, if U is any subset of Fn, x ∈ U is
an interior point of U if there exists r >
0 such that x ∈ B
⊆ U. In other
words U is an open set exactly when every point of U is an interior point of U.
If there is something called an open set, surely there should be something called a
closed set and here is the definition of one.
Definition 5.2.2 A subset, C, of Fn is called a closed set if Fn ∖ C is an open
set. They symbol, Fn∖C denotes everything in Fn which is not in C. It is also called
the complement of C. The symbol, SC is a short way of writing Fn ∖ S.
To illustrate this definition, consider the following picture.
You see in this picture how the edges are dotted. This is because an open
set, can not include the edges or the set would fail to be open. For example,
consider what would happen if you picked a point out on the edge of U in the
above picture. Every open ball centered at that point would have in it some
points which are outside U. Therefore, such a point would violate the above
definition. You also see the edges of B
dotted suggesting that
to be an open set. This is intuitively clear but does require a proof. This will
be done in the next theorem and will give examples of open sets. Also, you
can see that if
is close to the edge of U,
you might have to take r
to be very
It is roughly the case that open sets don’t have their skins while closed sets do. Here
is a picture of a closed set, C.
Note that x
and since Fn ∖ C
is open, there exists a ball, B
If you look at Fn ∖C,
what would be its skin? It can’t be in Fn ∖C
so it must be in C.
This is a rough heuristic explanation of what is going on with these
definitions. Also note that Fn
are both open and closed. Here is why. If x ∈∅,
there must be a ball centered at x
which is also contained in ∅.
This must be considered
to be true because there is nothing in ∅
so there can be no example to show it
Therefore, from the definition, it follows ∅
is open. It is also closed because if x
is also contained in
is both open and closed. From this,
it follows Fn
is also both open and closed.
Theorem 5.2.3 Let x ∈ Fn and let r ≥ 0. Then B
is an open set. Also,
is a closed set.
Proof: Suppose y ∈ B
It is necessary to show there exists r1 >
0 such that
Define r1 ≡ r −
it follows from the
above triangle inequality that
Note that if r
= 0 then B
the empty set. This is because if y ∈ Fn
and so y
has no points in it, it must be open because every
point in it, (There are none.) satisfies the desired property of being an interior
Now suppose y
and defining δ ≡
that if z ∈ B
then by the triangle inequality,
and this shows that B
⊆ Fn ∖ D
was an arbitrary point in
it follows Fn ∖D
is an open set which shows from the definition that
is a closed set as claimed.
A picture which is descriptive of the conclusion of the above theorem which also
implies the manner of proof is the following.