Finite Topology Concepts and Neighbourhoods

Let X be a general set. If a subset U(x) of X is determined for each element x in X, we call the pair (X, U(·)) a finite topological space. The subset U(x) is called a neighbourhood of x. In most cases, we assume x ∈ U(x) (in this case, we call X filled), but it is not necessary in all cases. U(x) is interpreted as a set of points neighbouring to x.

Using the notion of a neighbourhood, we can define many concepts.

Definition (Boundary). A boundary A^{∂} of a subset A of X is defined as:

A^{∂} = {x : U(x) ∩ A ≠ ∅ and U(x) ∩ A^{c} ≠ ∅},

where A^{c} is the complement of A and ∅ is the empty set.

In the field of image processing, the process of obtaining A^{∂} from A is called an extraction of the outline of A, which is shown in Figure 1.

We can divide A^{∂} into the following two parts – an inner boudary A^{∂_{i}} and an outer boundary A^{∂_{o}} which are defined as:

A^{∂_{i}} = A ∩ A^{∂} and A^{∂_{o}} = A^{c} ∩ A^{∂}.

Of course, we see from these definitions that A^{∂} = A^{∂_{i}} ∪ A^{∂_{o}}.

Definition (Interior). An interior A^{i} of a subset A of X is defined as:

A^{i} = {x : x ∈ A ⇒ U(x) ⊆ A}.

In image processing, the process of obtaining A^{i} is called a contraction of A.

Definition (Closure). A closure A^{b} of a subset A of X is defined as:

A^{b} = {x : U(x) ∩ A ≠ ∅}.

This concept is referred to as “expansion” in image processing.

Definition (Isolated Points). A set of isolated points A^{s} of a subset A of X is defined as:

A^{s} = {x : x ∈ A and (U(x) \ x) ∩ A = ∅}.

A point in A^{s} is referred to as “noise” in image processing. Conversely, we define a continuous part A^{n} of A as A^{n} = A \ A^{s}, which can also be written as:

A^{n} = {x : x ∈ A and (U(x) \ x) ∩ A ≠ ∅}.

A^{n} is called a “noise elimination” of A in image processing.

Definition (Inflation). An inflation A^{f} of a subset A of X is defined as:

A^{f} = {x : (∃y)(y ∈ A, x ∈ U(y))} = ⋃_{y∈A} U(y).

This concept seems similar to “closure”, but they do not always coincide. We present the following definition.

Definition (Symmetry). A neighbourhood U(·) of a finite topological space (X, U(·)) is called symmetric if and only if:

For all x, y ∈ X, y ∈ U(x) implies x ∈ U(y).

The neighbourhoods (a), (b), and (c) shown in Figure 2 are symmetric, but neighbourhoods (d), (e) and (f) are not.

By using the concept of symmetricity, we can give the following lemma.

Lemma 2.1. If a neighbourhood U(·) is symmetric, then for all subsets A:

A^{b} = A^{f}.

Proof. Let us look at the following formulae.

x ∈ A^{f} ⇔ for some y ∈ A, x ∈ U(y)

⇔ for some y ∈ A, y ∈ U(x) (by symmetricity) ⇔ U(x) ∩ A ≠ ∅

⇔ x ∈ A^{b}.

Thus, we get the result.

Q.E.D.

The following two definitions are analogies of very popular concepts of openness and closedness in continuous topologies.

Definition (Open Set). A subset G of X is open if and only if:

G = G^{i}.

Definition (Closed Set). A subset F of X is closed if and only if:

F = F^{b}.

We present here a second type of closure and interior.

Definition (f-closure and f-interior). An f-closure A^{f_{b}} and an f-interior A^{f_{i}} of a subset A of X is defined as:

A^{f_{b}} = ⋂{F : A ⊆ F, F is closed},
A^{f_{i}} = ⋃{G : G ⊆ A, G is open}.

We can show that an f-closure is gotten by a repetition of an ordinary closure (see Lemma 2.1 in Reference [1]) and an f-interior is gotten by a repetition of an ordinary interior.

Definition (Connected Set). A subset A of X is connected, if and only if, for any B, C in X:

A = B ∪ C, B ≠ ∅, C ≠ ∅, and B ∩ C = ∅ implies B^{b} ∩ C ≠ ∅.

This concept coincides with the usual intuitive concept of connectivity. In Figure 3, the image A is connected by the neighbourhood U_{9}, but not by the neighbourhood U_{5}.

For connectivity, we present the following theorem.

Theorem 2.1. Let X be filled and finite (i.e., containing only a finite number of points). Then, a subset A of X is connected if and only if for every x ∈ A:

(⋯ (({x}^{b} ∩ A)^{b} ∩ A)^{b} ⋯)^{b} ⊇ A.

That is, by a finite process of taking closures of A, A is covered. Proof. ⇒) We assume that for some x ∈ A:

(⋯ (({x}^{b} ∩ A)^{b} ∩ A)^{b} ⋯)^{b} ⊉ A.

For convenience, we denote an intersection of A and an n-th closure as P_{n} (= P_{n−1}^{b} ∩ A, P_{1} = {x}^{b} ∩ A). As X is finite, we can assume that P_{n+1} = P_{n}. If we let

B = P_{n}, C = A \ B (= A \ P_{n}),

then B ∪ C = B ∪ (A \ B) = A ∪ B = A and C ≠ ∅ (as P_{n} ⊉ A). Since X is filled, x ∈ P_{n}. Hence, x ∈ B = P_{n} ≠ ∅. B ∩ C = ∅ is clear. But,

B^{b} ∩ C = P_{n}^{b} ∩ C = P_{n} ∩ (A \ P_{n}) = ∅.

Therefore, X is not connected. ⇐) Let B and C be non-void subsets of A such that B ∩ C = ∅ and B^{b} ∩ C = ∅. Then, there exists an element x in B, and we can construct a set P_{n} as a procedure described previously and P_{n+1} = P_{n}.

Let us show that (B^{b} ∩ A)^{b} ∩ C = ∅. If the left hand side is not empty, there exists some element z in it such that:

z ∈ C and z ∈ (B^{b} ∩ A)^{b}.

Thus, U(z) ∩ B^{b} ∩ A ≠ ∅, where B^{b} ∩ A = B^{b} ∩ (B ∪ C) = (B^{b} ∩ B) ∪ (B^{b} ∩ C) = B ∪ B^{b} ∩ C. Hence, U(z) ∩ B^{b} ∩ C ≠ ∅ (as B ∩ U(z) = ∅ is clear), which contradicts with B^{b} ∩ C = ∅. Thus, we can set B = B^{b} ∩ A. If P_{i} ⊆ B^{b} ∩ A, then P_{i+1} ⊆ B^{b} ∪ A because:

P_{i+1} = P_{i}^{b} ∩ A ⊆ (B^{b} ∩ A)^{b} ∩ A = B^{b} ∩ A.

Therefore, we obtain the result:

P_{n} ⊆ B^{b} ∩ A = B^{b} ∩ (B ∪ C) = B,

i.e.,

P_{n} ⊉ A.

Q.E.D.

The following facts are easily derived:

1. ((A^{c})^{i})^{c} = A^{b}, ((A^{c})^{b})^{c} = A^{i}.

2. A^{∂} = A^{b} ∩ (A^{c})^{b}, (A^{c})^{∂} = A^{∂}.

3. If x ∈ A^{s}, then X ∉ (A \ {x})^{b}.

4. If A^{s} ≠ ∅, then A is not connected.

5. If X is filled, then A ⊆ A^{b}, A^{i} ⊆ A.

6. If A is open, then A^{c} is closed. Conversely, if A is closed, then A^{c} is open.