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Let the set be endowed with the Euclidean distance
, where
and
.
If and
, the set
is called the
-disk about
, or
-ball about
, or
-neigborhood of
. A set
is called open if for each
, there exists an
such that
.
Example: If is open and
, then
is also open.
A point is called an interior point of
if there is an open set
such that
. The set
of all interior points of
is called the interior of
.
The interior of a set may be empty. On the other hand, the interior is the largest subset of
.
A set is called closed if its complement
is open.
A point is called an accumulation point (or cluster point) of a set
if every open set
containing
contains some point of
other than
.
A set is closed if and only if all accumulation points of
belong to
.
The closure of a set
is defined as the intersection of all closed sets containing
. The closure of
is the union of
and the set of all accumulation points of
.
If then the boundary
(sometimes also denoted by
) is defined as the set
.
We have if and only if for every
, the
-neigborhood
of
contains points of
and of
.
A set is bounded if there is a positive real number
such that
for all
.
A set is called totally bounded if for each
there is a finite set
of points in
such that
.
A totally bounded set is bounded.
A cover of is a collection
od sets whose union contains
. In this case we also say that
covers
. A cover is called open if each
is open. A subcover of a given cover is a subcollection of
whose union also contains
. A finite subcover is the subcollection containing only a finite number of sets.
A subset is called compact if every open cover of
has a finite subcover. A compact set is closed.
Heine-Borel Theorem: A set is compact if and only if it is closed and bounded.
A collection of closed sets in
is said to have the finite intersection property for an
if the intersection of any finite number of the
with
is non-empty.
A set is compact if and only if every collection of closed sets with the finite intersection property for
has non-empty intersection with
.
Nested Set Property: If is a sequence of compact non-empty sets in
such that
then the intersection
is non-empty.
If a set is called convex if it has the property that for each pair of points belonging to it the straight line segment connecting them consists entirely of points which also belong to the set.
An open set is called path-connected in case each pair of points in
can be joined by a path consisting of a finite number of straight line segments joined end to end consecutively, the whole path lying entirely in
and not crossing itself anywhere. Such a path is called a polygonal arc.
Let . Two open sets
are said to separate
if they satisfy these conditions:
The set is called disconnected if such sets exists, and if such sets do not exist, we say that
is connected.
A set is called totally disconnected if
and
then
and
where
and
are open sets that disconnect
.
Path-connected sets are connected. On the other hand, if is an open connected subset of
then
path-connected.
Maximal connected subsets of a set are called components. Similarly we can define path components using path-connectedness.
A set is called star-shaped (or star convex) if its boundary is entirely visible from an interior point or from a point on the boundary. It is called strictly star-shaped if this is valid when the visibility is restricted to the interior of the set.
Clearly a set is convex it is star shaped with respect to each of its points.
A set is called symmetric with respect to a point
if for every
also
. The point
is called the centre of symmetry of this set.
Cite this web-page as:
Štefan Porubský: Point Sets in a Euclidean Space.