Mathematical Morphology and Rough Sets - Rough Fuzzy Image Analysis: Foundations and Methodologies - page 67

section. Those readers who are familiar with these concepts, may ignore this section. The
main reference of this section is the topic written by M.C. Gemignani,(Gemignani, 1990).

Topology: Let X be a non-empty set. A collection τ of subsets of X is said to be a

topology on X if
•X and φ belongs to τ
•The union of any finite or infinite number of sets in τ belongs to τ
•The intersection of any two sets in τ belongs to τ .

The pair (X, τ ) is called a topological space. Several topologies can be defined on every set
X.

Discrete Topology: if (X, τ )is a topological space such that, for every x∈X, the single-
ton set{x}is in τ , then τ is the discrete topology.

Open and closed sets are pivotal concepts in topology.
Open sets: Let (X, τ ) be a topology. Then the members of τ are called open sets. There-
fore,

•X and φ are open sets.
•The union of any finite or infinite number of open sets are open sets.
•The intersection of any finite number of open sets is an open set.

Closed Sets: Let (x, τ ) be a topological space. A set S⊆X is said to be closed, if X\S
is open.
•φ and X are closed sets.
•The intersection of any finite or infinite number of closed sets is a closed set.
•The union of any finite number of closed set is a closed set.

Some subsets of X, may be both closed and open. In a discrete space, every set is both
open and closed, while in a non-discrete space all subsets of X are neither open nor closed,
except X and φ.

Clopen Sets: A subset S of a topological space (X, τ ) is said to be clopen if it is both
closed and open in (X, τ ).

The concept of limit points are closely related to topological closure of a set.
Limit Points: Let A be a subset of a topological space (X, τ ). A point x∈X is said to
be a limit point (cluster point or accumulation point)of A, if every open set, O containing
x contains a point of A different from x.
The following propositions provide a way of testing a set to determine if it is closed or not.

Proposition 4.2.1 Let A be a subset of a topological space (X, τ ). Then A is closed if
it contains all of its limit points.

Proposition 4.2.2 Let A be a subset of a topological space (X, τ ), and A 0 be the set of
all limit point of A, then A∪A 0 is closed.

The topological concepts of closure and interior play an important role in this chapter. A
brief explanation of these concepts is given next.