SEQUENTIALLY HAUSDORFF AND FULL SEQUENTIALLY HAUSDORFF SPACES

In this paper, we define the notions of sequentially Hausdorff Space and full sequentially Hausdorff space. Also we give the relationships between these notions and Hausdorffness.


Introduction
Hausdor¤ness is one of the most important properties of topological spaces.A number of topological spaces encountered are Hausdor¤.In a Hausdor¤ space, sequential limits are unique [1,3,4].Sequences are useful for studying functions, spaces, and other mathematical structures using the convergence properties of sequences.
On the other hand, in topology and related …elds of mathematics, a sequential space is a topological space that satis…es a very feeble axiom of countability.Every open set in a topological space is also a sequentially open set [5,6,7].A sequential space is a space in which every sequentially open set is open.Every …rst countable space is a sequential space.
The notion of sequentially continuity is wider than the notion of continuity that is; every continuous function is sequential continuous.In general topology the notions of sequentially continuity, sequentially compactly and sequentially connectedness play important roles.Despite the fact that in …rst countable spaces these notions are equal to the notions of continuity, compactly and connectedness, respectively, there is no any generalization for other spaces.The relations between these notions give us some useful results.They are explained by using sequentially open and sequentially closed sets [8,9,10,12].We also refer to [13,14].The concept of G-sequential compactness was introduced in [9].On the other hand the notion of G-sequential continuity was investigated in [11] and the de…nition of sequential SEQUENTIALLY HAUSDORFF AND FULL SEQUENTIALLY HAUSDORFF SPACES 1725 connectedness was given in [10].Also in [15], Mucuk and Şahan gave a further investigation of G-sequential continuity in topological groups.
In this study, we …rst de…ne the notion of sequentially Hausdor¤ spaces and give some related properties and examples.We explain the relationship between Hausdorfness and sequentially Hausdorfness on topological spaces.We also de…ne full sequentially Hausdor¤ spaces and give the relationships with both Hausdor¤ spaces and sequentially Hausdor¤ spaces.

Preliminaries
In a Hausdor¤ space, given any two distinct points, there are disjoint open sets containing the two points respectively.De…nition 2.1.[1,2,3,4] Let X = (X; ) be a topological space, (x n ) a sequence in X and x 2 X.If, for every open neighborhood G of x, there is a natural number n 0 such that x n 2 G whenever n n 0 , then we say that the sequence (x n ) converges to x.
In other words, the sequence De…nition 2.2.[13]Let X be a topological space and let A be a subset of X.We say that A is sequentially open if each sequence in X converging to a point in A is eventually in A; that is, if whenever (x n ) is a sequence in X that converges to a point in A, then there exists a natural number n 0 such that x n belongs to A whenever n n 0 .De…nition 2.3.[13]Let X be a topological space and let F be a subset of X. F is called sequentially closed if no sequence in F converges to a point not in F .
By the De…nition 2.2 and De…nition 2.3, it is easy to see that in a topological space every open set is sequentially open and every closed set is sequentially closed.Proposition 2.4.[13]Let X be a topological space and A X. Then A is sequentially open if and only if XnA is sequentially closed.
Hence we observe the following proposition: Proposition 2.5.[13]Let X be a topological space.Then the followings are equivalent: Each subset of X which intersects every convergent sequence in a closed set is closed, (iv) Each subset of X which intersects every compact metric subspace of X in a closed set is closed.
De…nition 2.6.[13]Let X be a topological space.Then we say X is a sequential space or sequential if every sequentially open subset of X is open.
[13]Let X be a set and is the discrete topology on X.Then the topological space (X; ) is a sequential space.
Example 2.8.[13]Let X be an uncountable set and is the co-countable topology on X.Then the topological space (X; ) is not a sequential space.
Proposition 2.9.[13] Each subspace of a sequential space is sequential.
De…nition 2.10.[5,6,7]Let X and Y be topological spaces, f : X !Y be a function and x 2 X.If for each sequence (x n ) converging to the point x, the sequence

Sequentially Haussdorf Spaces
In this section, we de…ne sequentially Haussdor¤ spaces and give some properties by using the notions of sequentially open set and sequentially closed set.On the contrary, the following example shows us that every sequentially Haus-dor¤ space does not have to be a Hausdor¤ space.
Example 3.7.Let R be the set of all real numbers and be the co-countable topology on R. We know that (X; ) is a sequentially Haussdorf space but not a Hausdor¤ space.Proposition 3.8.If (X; ) is sequentially Hausdor¤ , then sequential limits in X are unique, that is whenever (x n ) converges to x and y, then x = y.
Proof: Suppose that (X; ) be sequentially Hausdor¤.Let (x n ) be a sequence in X converging to some point x and y and let x 6 = y.Then there is a disjoint sequential open sets G and H such that x 2 G and y 2 H. Since G is sequentially open, each sequence (x n ) in X converging to x is eventually in G; that is, there exists a natural number n 1 such that x n 2 G whenever n n 1 .On the other hand since H is sequentially open, each sequence (x n ) in X converging to y is eventually in H; that is, there exists a natural number n 2 such that x n 2 H whenever n n 2 .If we choose n 0 as maximum of the numbers n 1 and n 2 , then there exists a natural number n n 0 we obtain that x n 2 G \ H.But we know that G \ H = ;.So sequential limits in X are unique.Proposition 3.9.Let (X; ) be a sequential space.Then (X; ) is sequentially Hausdor¤ if and only if (X; ) is Hausdor¤ .Proposition 3.10.Let (X; ) be a sequentially Hausdor¤ space and A X. Then the subspace topology A on A is sequentially Hausdor¤ .
Proof: The proof is clear.Proof: Let (X; ) be sequentially Hausdor¤ and distinct points y 1 ; y 2 2 Y .Then there exist two distinct points x 1 ; x 2 2 X such that f (x 1 ) = y 1 and f (x 2 ) = y 2 , since the function is one-to-one and onto.Hence, there exist sequentially open subsets x 1 2 G 1 and x 2 2 G 2 such that G 1 \ G 2 = ;.For the sequences (x n ) and (x 0 n ) converging to x 1 and x 2 are eventually in G 1 and G 2 , respectively.Since f is continuous, so it is sequentially continuous.Then we have that the sequences f (x n ) and f (x 0 n ) converge to f (x 1 ) and f (x 2 ), respectively.So they are eventually in sequential open sets f (G 1 ) and f (G 2 ), respectively.Since f (G 1 ) \ f (G 2 ) = ;, the proof is completed.Corollary 3.12.Sequentially Hausdor¤ ness of a space is a topological property.
Proof: Let (X; ) and (Y; ) be topological spaces and the function f : (X; ) ! (Y; ) be a homeomorphism; that is, f be one-to-one, onto and continuous and the inverse function f 1 be continuous.The conditions of the Proposition 3.11 are su¢ cient to preserve the structure of sequentially Hausdorfness.De…nition 3.13.Let X be an in…nite set and (X; ) be a topological space.Let for any two distinct points x; y 2 X, there exist sequentially open sets G 3 x and H 3 y such that G \ H = ;.If there exist sequences x n and y n , whose terms are several and converging to x and y, respectively, such that x n has no term in H and also y n has no term in G, then (X; ) is called a full sequentially Hausdor¤ space.Proposition 3.14.Every full sequentially Hausdor¤ space is sequentially Haus-dor¤ .
Proof: By the De…nition 3.13, for any two distinct points x; y 2 X, if there are two distinct sequentially open sets G 3 x and H 3 y such that G \ H = ;, then (X; ) is a sequentially Hausdor¤ space.
Example 3.15.The discrete topology on a set X is full sequentially Hausdor¤ .
Example 3.16.Let X 6 = ; be a set and be the co-countable topology on X.Then (X; ) is full sequentially Hausdor¤ .
Example 3.17.Let us consider the usual topology U on R and distinct points x, y and x < y.Let the distance between the points x and y be .Then for x and y, there exists disjoint sequentially open sets (x 3 ; x + 3 ) and (y 3 ; y + 3 ), respectively.For a chosen sequence (x n ) = (x 1 n ), it has no term in (y 3 ; y + 3 ).Similarly, for a chosen sequence (y n ) = (y + 1 n ), it has no term in (x 3 ; x + 3 ).So the usual topology on R is full sequentially Hausdor¤ .Proof: If (X; ) is Hausdor¤, then given any two distinct points x; y in X, there exists sequentially open subsets G 3 x and H 3 y such that G \ H = ;.Since G is sequentially closed, then there exist a sequence (x n ) converging to x is in G exactly.So H does not include any term of (x n ).Since H is sequentially closed, then there exists a sequence (y n ) converging to y is in H exactly.So G does not include any term of (y n ).
Proof: We know that every full sequentially Hausdor¤ space is sequentially Hausdor¤.By the Proposition 3.8, if (X; ) is sequentially Hausdor¤, then sequential limits in X are unique.Proof: Chosen distinct points y 1 ; y 2 2 Y , since the function f is one-to-one and onto, there exist two distinct points x 1 ; x 2 2 X such that f (x 1 ) = y 1 and f (x 2 ) = y 2 .By the proposition 3.11, we know that (Y; ) is sequentially Hausdor¤; that is, there exist sequentially open subsets x 1 2 G 1 and x 2 2 G 2 such that G 1 \ G 2 = ;.Since (X; ) is full sequentially Hausdor¤, then the sequences (x n ) and (x 0 n ) converging to x 1 and x 2 have no element in H and G, respectively.So the sequences (f (x n )) and (f (x 0 n )) converging to f (x 1 ) and f (x 2 ) have no element in f (H) and f (G), respectively, where f (H) 3 f (x 1 ) and f (G) 3 f (x 2 ) are sequentially open.So (Y; ) is also full sequentially Hausdor¤.

Sequentially Compact Spaces
In this section we give a new de…nition of sequentially compact spaces which are de…ned by sequentially open sets.

De…nition 3 . 1 .
A topological space X is called a sequentially Haussdorf space or sequentially Haussdorf if given any two distinct points x; y in X, there exist sequentially open subsets G 3 x and H 3 y such that G \ H = ;.Example 3.2.Since every open set is sequentially open, the discrete topology on a set is sequentially Haussdorf.Example 3.3.Let X 6 = ; be a set and be the co-countable topology on X.Then every subset of X is sequentially open.Let x; y 2 X are distinct points and G = fxg.Since y 2 G c and G \ G c = ;, then (X; ) is sequentially Hausdor¤ .Example 3.4.Let us consider the topology = fG a = (a; 1) ja 2 Rg [ f;g on R for every a 2 R. Then the topological space (R; ) is not sequentially Hausdor¤ .Example 3.5.Every …rst countable space is sequentially Hausdor¤ .The notion of sequentially Hausdor¤ space is wider than the notion of Hausdor¤ space.We now give the following proposition: Proposition 3.6.Every Hausdor¤ space is sequentially Hausdor¤ .Proof: Let (X; ) be a topological space.Since any two distinct points x; y in X, there exist open subsets G 3 x and H 3 y such that G \ H = ; and also every open set is sequentially open, then (X; ) is sequentially Hausdor¤.

Proposition 3 .
21. Let R be the set of real numbers and d(x; y) = jx yj be the usual metric on R. Then (R; d) is full sequentially Hausdor¤ .Proof: Let x; y 2 X and d(x; y) = 1.Chosen open discs G = D(x; 1 3 ) and H = D(y; 1 3 ) are sequentially open neighborhoods of x and y, respectively.It is easy to see that G \ H = ;.Then for the sequence (x n ) = (x + 2 3n ) converging to x, it has no term in H. Similarly for the sequence (y n ) = (y + 2 3n ) converging to y, it has no term in G. Then (X; d) is full sequentially Hausdor¤.Proposition 3.22.Let (X; ) and (Y; ) be topological spaces and let the function f : (X; ) ! (Y; ) be sequentially open, one-to-one, onto and continuous.If (X; ) is full sequential Hausdor¤ then (Y; ) is full sequential Hausdor¤ .

De…nition 4 . 1 .De…nition 4 . 2 .Proposition 4 . 3 .
Let X be a topological space and G = fG i ji 2 Ig be a class of any sequentially open subsets of X. IfX [ i2I G i ;then G is called a sequentially open cover of X.Let X be a topological space.If each of sequentially open covers of X has a …nite subcover, then X is called a sequentially compact space.Every sequentially compact space is compact.Proof: Let X be a sequentially compact space and the class G = fG i ji 2 Ig be an open cover of X.Since every open set is sequentially open, 8G i 2 G is sequentially open.So the class G is a sequentially open cover of X.Since X is sequentially compact, G has a …nite subcover.Also it means that if G = fG i ji 2 Ig be an open cover of X, then G has a a …nite subcover.So X is compact.

Example 4 . 4 .Example 4 . 5 .
Let (X; d) be a metric space where X is unbounded.Then X is not sequentially compact.Because the class of all sequentially open sets D(a; 1), for every a 2 X, is a sequentially open cover of X, but does not have any …nite subcover.Every …nite set is sequentially compact.