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Geometry
Combinatorial Geometry
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Main Index
Geometry
Combinatorial Geometry
Subject Index
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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.