# Minimal sets examples of thesis

theme_minimal. A minimalistic theme with no background annotations. theme_classic. A classic-looking theme, with x and y axis lines and no gridlines. theme_void. A completely empty theme. theme_test. A theme for visual unit tests. It should ideally never change except for new features. Examples. It is called a strongly minimal set if this is true even in all elementary extensions. A strongly minimal set, equipped with the closure operator given by algebraic closure in the model-theoretic sense, is an infinite matroid, or pregeometry. A model of a strongly minimal theory is determined up to isomorphism by its dimension as a matroid. This chapter presents examples of exceptional minimal sets. A codimension one foliation is transversely-PL k + (IR) if all holonomy transitions are restrictions of elements of PL k + (IR) A codimension one foliation is without one-sided holonomy if for every loop in a leaf, the associated holonomy germ is either trivial or nontrivial on both pomononslici.cf by: 6. A characterization of all bounded, closed and convex sets in R n that are invariant K-minimal for (ℓ 1, ℓ ∞) is established. Paper II presents examples of invariant K-minimal sets in R n for (ℓ 1, ℓ ∞). A convergent algorithm for computing the element with minimal K-functional in such sets is pomononslici.cf by: 1. In this paper, we introduce and define minimal-open sets in topological spaces and we obtain some basic properties of this set. Moreover, we define-locally finite space and give some applications for finite minimal-open sets.

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To learn more, view our Privacy Policy. Log In Sign Up. Alias B. Khalaf1, Halgwrd M. In this paper, we introduce and define minimal -open sets in topological spaces and we obtain some basic properties of this set. Moreover, we define -locally finite space and give some applications for finite minimal -open sets. Introduction The study of semi open sets in topological space was initiated by Levine [4]. In [1], Namiq defined an operation on the family of semi open sets in a topological space called s-operation by using this operation he defined -open sets which is equivalent to - open set defined in [2].

By using s-operation and semi closed sets Namiq in [3], defined - open set and also investigated several properties of -derived, -interior and -closure points in topological spaces. In this paper, we introduce and discuss minimal -open sets in topological spaces.

We establish some basic properties of minimal -open sets. Throughout the present paper or simply denote a topological space or simply space.

## Minimal pair

Preliminaries First, we recall some definitions and results used in this paper. For any subset of then the closure and the interior of are denoted by and respectively. The family of all semi open resp. We consider as a function defined on into the power set of , and is called an s-operation if for each nonempty semi open set It is assumed that and for any s-operation Let be a space and be an s-operation, then a subset of is called a -open set [1]which is equivalent to -open set[2] if for each there exists a semi open set such that and The complement of a -open set is said to be -closed.

The family of all -open resp. Definition 2.

A -open subset of a space is called -open [3] if for each there exists a semi closed set such that The complement of a -open set is said to be -closed[3]. Proposition 2. For a space The following examples show that the converse of the above proposition may not be true in general. Example 2. Let be a space, an s-operation is said to be s-regular. If for every semi open sets and of there exists a semi open set containing x such that Definition 2.

Let be a space and let be a subset of Then: 1 The -closure of is the intersection of all -closed sets containing 2 The -interior of is the union of all -open sets of contained in Proposition 2. For each point if and only if for every such that Proposition 2. Let be a s-regular s-operation. If and are -open sets in Then is also a -open set. Minimal -open sets Definition 3. Let be a -open subset of a space.

Then is called a minimal - open set if and are the only -open subsets of Example 3. Let and We define an s-operation as if and otherwise. Then the -open sets are Here is minimal -open set. Proposition 3. Let be a topological space and suppose that is a —regular operation.

Then: 1 Let be a minimal -open set and a -open set. Then or.. Then or. Therefore Proposition 3. Let be -regular and be a minimal -open set. If is an element of then for any -open neighborhood of.

Let be a -open neighborhood of such that B. Since is a -regular operation, then is -open set such that and This contradicts our assumption that is a minimal -open set. Let be a minimal -open set. Then for any element of is -open neighborhood of , where is -regular.

By Proposition 3. Let be a minimal -open set in and such that Then for any -open neighborhood of or where is -regular. Since is a -open set, then the results is followed by Proposition 3.

Corollary 3. Let be a minimal -open set in and such that If is -open neighborhood of Then or where is -regular. If for any -open neighborhood of then is -open neighborhood of Therefore Otherwise there exists a -open neighborhood of such that Then we have Corollary 3.

## Strongly minimal theory

If is a nonempty minimal -open set of and is -regular. Let be any nonempty subset of Let and be any -open neighborhood of By Proposition 3. Let be a non-empty -open subset of a space If then for any non-empty subset of Proof. For any non-empty subset of we have On the other hand, by supposition we have implies Proposition 3. Let be a non-empty -open subset of a space If for any non-empty subset of then is a minimal -open set.

Suppose that is not a minimal -open set. Then there exists a non-empty - open set such that and hence there exists an element such that Then we have This implies that which is impossible. Hence is a minimal -open.

Combining Corollary 3. Let be -regular non-empty -open subset of the following are equivalent: 1 is minimal -open set. Finite -open sets In this section, we study some properties of minimal -open sets in finite -open sets and -locally finite spaces.

## Complete themes

Proposition 4. Let be a space and a finite -open set in Then there exists at least one non-empty finite minimal -open set such that Proof. Suppose that is a finite -open set in Then we have the following two possibilities: 1 is a minimal -open set.

In case 1 , if we choose then the proposition is proved. If the case 2 is true, then there exists a non-empty finite -open set which is properly contained in If is minimal -open, we take If is not a minimal -open set, then there exists a non-empty finite -open set such that We continue this process and have a sequence of -open sets Since is a finite, this process will end in a finite number of steps.

That is, for some natural number we have a minimal -open set such that This completes the proof. Definition 4. A space is said to be a -locally finite space, if for each there exists a finite -open set in such that Proposition 4.

Let be a -locally finite space and a non-empty -open set. Then there exists at least one finite minimal -open set such that where is - regular. Since is a non-empty set, there exists an element of Since is a - locally finite space, we have a finite -open set such that Since is a finite -open set, we get a minimal -open set such that by Proposition 4.

Let be a space and for any a -open set and a finite -open set. Then is a finite -open set, where is -regular. We see that there exists an integer n such that and hence we have the result. Using Proposition 4. Let be a space and for any a -open set and for any a non-empty finite -open set. Applications Let be a non-empty finite -open set. It is clear, by Proposition 3.

Let be a space and a finite -open set such that is not a minimal -open set.

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Let be a class of all minimal -open sets in and Define is -open neighborhood of. Then there exists a natural number such that is contained in where is -regular. Suppose on the contrary that for any natural number is not contained in By Corollary 3. Therefore by Proposition 4. Hence the result followed. Corollary 5. Let be a space and be a finite -open set which is not a minimal -open set.

Let be a class of all minimal -open sets in and Then there exists a natural number such that for any -open neighborhood of is contained in where is -regular. This follows from Theorem 5. Let be the class of all minimal -open sets in and Then there exists a natural number such that where is -regular.

It follows from Corollary 5. Let be a finite -open set in a space and for each is a minimal -open sets in A. If the class contains all minimal -open sets in then for any where is -regular. If is a minimal -open set, then the result is followed by Theorem 3.