This paper introduces the concept of almost separable spaces, a generalization of classical separable spaces in topology. We investigate the fundamental properties of almost dense subsets, establishing their relationships with dense sets, sequential separability, and strong sequential separability. We demonstrate that almost separability is c-productive, and under certain conditions, the converse holds for infinite products. Furthermore, we analyze the cardinality of the set of real-valued continuous functions on almost separable spaces and provide bounds for functionally Hausdorff spaces. Finally, we extend the classical Baire category theorem to pseudocompact spaces using almost dense cozero sets, highlighting implications for connectedness and topological structure. Several examples illustrate these properties in noncompletely regular, normal, and functionally Hausdorff spaces
Let Xbe any topological space, and let C(X)denote the set of all real-valued continuous functions on X. A subset A⊂Xis called almost dense if, for every f∈C(X), the condition f(A)={0}implies f(X)={0}. A topological space Xis said to be almost separable if it contains a countable almost dense subset.
It is clear that every dense subset is automatically almost dense. In the context of completely regular spaces, the notions of dense and almost dense coincide; however, the converse does not necessarily hold in general topological spaces. For instance, there exist non-completely regular spaces where dense and almost dense subsets coincide, illustrating the subtle differences between these concepts. In this study, we also show that almost separability is c-productive, and under certain conditions, the converse holds as well. Additionally, we explore the relationships between almost separability, sequential separability, and strongly sequential separability.
Definition 1.1 ([1,3,4]). A topological space Xis sequentially separable if there exists a countable subset D⊂Xsuch that, for every x∈X, there exists a sequence from Dconverging to x.
Definition 1.2 ([1]). A space Xis called strongly sequentially separable if it is separable and every countable dense subset is sequentially dense. A subset D⊂Xis sequentially dense if, for every x∈X, there exists a sequence in Dthat converges to x.
1. Almost Dense Subsets of a Topological Space
Definition 2.1. A subset A⊂Xis almost dense if, for all f∈C(X), the equality f(A)={0}implies f(X)={0}.
Definition 2.2. A subset A⊂Xis called a zero set if there exists f∈C(X)such that A=Z(f)={x∈X:f(x)=0}. The complement of a zero set is referred to as a cozero set.
Theorem 2.3. Every dense subset of a topological space is almost dense.
Proof. Let Abe a dense subset of X. Consider any f∈C(X)such that f(A)={0}. By continuity and the density of A, f(X)={0}. Therefore, Ais almost dense. □
Theorem 2.4. In a completely regular space X, every almost dense subset is dense.
Proof. Suppose Ais almost dense but not dense in X. Then there exists x_0∈X∖A. By the property of complete regularity, there exists f∈C(X)such that f(x_0)=1and f(A)={0}. This contradicts the assumption that Ais almost dense. Hence Amust be dense in X. □
Theorem 2.5. Let Ybe a dense subset of X(which may not be completely regular) such that the relative topology of Yhas a base consisting of cozero sets of X. Then every almost dense subset of Xis dense.
Proof. Let Abe almost dense in X, and let Ube a non-empty open subset of X. Since Yis dense, U∩Y≠∅. If A∩U=∅, then A∩U∩Y=∅. Choose y∈U∩Y. By the cozero base property, there exists f∈C(X)such that y∉Z(f)⊂U∩Y. Then f(A)={0}, but f(X)≠{0}, contradicting the almost denseness of A. Therefore, A∩U≠∅. □
Example 2.6. Consider the K-topology on R, defined as β={(a,b):a<b,a,b∈R}∪{(a,b)∖K:a<b,a,b∈R}with K={1/n:n∈N}. The space (R,τ_K)is not regular, hence not completely regular. Let Y=R∖{0}. The subspace Yis open, dense, and completely regular in the relative topology with a cozero base. Applying Theorem 2.5, any almost dense set in (R,τ_K)is dense.
Example 2.7. Let X={a,b}with topology τ={∅,X,{a}}. Then Xis normal. The subset {b}is almost dense but not dense in X, showing that almost dense sets may not be dense in normal spaces.
Theorem 2.8. Let f:X→Ybe a continuous, surjective map. If A⊂Xis almost dense, then f(A)is almost dense in Y.
Proof. Suppose g∈C(Y)with g(f(A))={0}. Then g∘f∈C(X)satisfies g∘f(A)={0}. Since Ais almost dense, g∘f(X)={0}. Surjectivity of fimplies g(Y)={0}, proving that f(A)is almost dense in Y. □
Theorem 2.9. Let τ_1and τ_2be two topologies on Xwith τ_2finer than τ_1. If Ais almost dense in (X,τ_2), then it is almost dense in (X,τ_1).
Proof. For any f∈C(X,τ_1), since τ_2is finer, f∈C(X,τ_2). By the almost denseness of Ain (X,τ_2), we have f(X)={0}. □
Theorem 2.10. If A⊂Xis almost dense and B⊂Yis almost dense, then A×B⊂X×Yis almost dense.
Proof. Let f:X×Y→Rbe continuous and vanish on A×B. Fix a∈Aand define f_a (y)=f(a,y). Then f_a (B)={0}, so f_a (Y)={0}. Similarly, for y∈Y, define f_y (x)=f(x,y). Then f_y (A)={0}, so f_y (X)={0}. Therefore, f=0on X×Y, proving that A×Bis almost dense. □
Corollaries and Propositions on Product Spaces
Corollary 2.11. Let A_1,A_2,…,A_nbe almost dense subsets of X_1,X_2,…,X_n, respectively. Then their Cartesian product ∏_(i=1)^n A_iis almost dense in ∏_(i=1)^n X_i.
Proposition 2.12. Let {X_α:α∈Λ}be a family of topological spaces, and for each α∈Λ, let A_α⊂X_αbe almost dense. Then the product ∏_(α∈Λ) A_αis almost dense in ∏_(α∈Λ) X_α.
Proof. Fix an element a=(a_α )_(α∈Λ)∈∏_(α∈Λ) A_α, and define
D={(x_α )_(α∈Λ)∈∏X_α:{α:x_α≠a_α}" is finite"}.
Let f:∏_(α∈Λ) X_α→Rbe continuous such that f=0on ∏_(α∈Λ) A_α. For any finite subset I⊂Λ, the product ∏_(α∈I) A_αis almost dense in ∏_(α∈I) X_αby Corollary 2.11. Consequently, the subset ∏_(α∈I) A_α×{a_α:α∈Λ∖I}is almost dense in ∏_(α∈I) X_α×{a_α:α∈Λ∖I}. This ensures that f(D)={0}, completing the proof. □
Theorem 2.13. Let {Y_α:α∈Λ}be a family of topological spaces and fix a=(x_α )_(α∈Λ)∈∏_(α∈Λ) Y_α. Then the set
E={(y_α )_(α∈Λ)∈∏_(α∈Λ) Y_α:y_α≠x_α " for finitely many " α}
is dense in ∏_(α∈Λ) X_α. Hence, if f=0on E, then f=0on ∏_(α∈Λ) X_α, establishing that ∏_(α∈Λ) A_αis almost dense in the product space. □
Connectedness and Almost Dense Sets
Theorem 2.14. If a topological space Xcontains a connected almost dense subset, then Xis connected.
Proof. Let A⊂Xbe connected and almost dense. Assume, for contradiction, that Xis disconnected. Then there exists a continuous surjection f:X→{0,1}. Let Y=f^(-1) (0), so that X∖Y=f^(-1) (1). Since Ais connected, either A⊂Yor A⊂X∖Y.
Case 1: If A⊂Y, then f(A)={0}but f(X)≠{0}, contradicting the almost denseness of A.
Case 2: If A⊂X∖Y, define g=1-f. Then g(A)={0}but g(X)≠{0}, again contradicting almost denseness.
Hence, Xmust be connected. □
Remark. The converse is not true; a connected space can contain disconnected almost dense subsets. For example, let X=Rwith the topology of all subsets containing 0. Then Xis connected, but the subset A={1,2}is almost dense and disconnected.
Characterization of Almost Dense Sets via Cozero Sets
Theorem 2.15. A subset A⊂Xis almost dense if and only if it intersects every nonempty cozero set in X.
Proof. Suppose Ais almost dense and let U=X∖Z(f)be a nonempty cozero set. If A∩U=∅, then f(A)={0}. By almost denseness, f(X)={0}, contradicting U≠∅. Conversely, assume Aintersects every nonempty cozero set. If Awere not almost dense, there would exist f∈C(X)with f(A)={0}but f(X)≠{0}. Then X∖Z(f)is a nonempty cozero set not intersecting A, a contradiction. □
Remark. This result is analogous to the classical property of dense sets, which intersect every nonempty open set. It naturally leads to the question: if Ais almost dense in Xand U⊂Xis a nonempty cozero set, is A∩Ualmost dense in U? The answer is not straightforward and requires further investigation.
Almost Separable Spaces
Definition 3.1. A topological space Xis almost separable if it contains a countable almost dense subset.
Theorem 3.2. Every separable space is almost separable.
Remark. The converse is false, as illustrated below.
Example 3.3. Let X=Rwith the cocountable topology τ_c. Any countable subset Ais not dense, since X∖Ais open and A∩(X∖A)=∅. However, Q, the set of rationals, is almost dense. Any continuous f:X→Ris constant, so if f(Q)={0}, then f(X)={0}. Therefore, Xcontains a countable almost dense subset and is almost separable.
Theorem 3.4. The finite product of almost separable spaces is almost separable.
Proof. Immediate from Corollary 2.11. □
Theorem 3.5. Let Y⊂Xbe almost dense and almost separable as a subspace. Then Xis almost separable.
Proof. Let Abe a countable almost dense subset of Y. For any f∈C(X)with f(A)={0}, the restriction f∣_Y∈C(Y)satisfies f∣_Y (A)={0}. Since Ais almost dense in Y, f∣_Y (Y)={0}. By the almost denseness of Yin X, f(X)={0}. Therefore, Ais a countable almost dense subset of X. □
Theorem 3.6. Let {X_α:α∈Λ}be a family of almost separable spaces with ∣Λ∣=c. Then ∏_(α∈Λ) X_αis almost separable.
Proof. Let A_α⊂X_αbe countable almost dense subsets. Define bijections f_α:N→A_αand construct f_Λ:N^Λ→∏_(α∈Λ) A_αby f_Λ ((n_α )_(α∈Λ))=(f_α (n_α))_(α∈Λ). This mapping is continuous and surjective. Since N^Λis separable (hence almost separable), ∏_(α∈Λ) A_αis almost separable. As it is almost dense in the full product, the product space is almost separable. □
Pseudocompactness and Baire Category–Type Theorem
Theorem 4.1. For any topological space X, the following statements are equivalent:
Xis pseudocompact, meaning every real-valued continuous function on Xis bounded.
If {F_n:n∈N}is a sequence of zero sets in Xhaving the finite intersection property, then ⋂_(n=1)^∞ F_n≠∅.
If {U_n:n∈N}is a countable collection of cozero sets covering X, then a finite subcollection suffices to cover X.
This theorem provides multiple characterizations of pseudocompact spaces, connecting boundedness of continuous functions with the intersection behavior of zero sets and the covering properties of cozero sets.
Theorem 4.2. Let Xbe a topological space. For any nonempty cozero set Uand any point x∈U, there exist a cozero set Vand a zero set Fsuch that
x∈V⊆F⊂U.
Proof. Since Uis cozero, there exists a continuous function f:X→[0,1]such that U=f^(-1) ((0,1])=X∖f^(-1) ({0}). Because x∈U, we have f(x)>0. Choose a small δ>0so that 0<f(x)-δ<f(x). Define
V=f^(-1) ((f(x)-δ,1]),F=f^(-1) ([f(x)-δ,1]).
Clearly, x∈V⊆F⊂U. Here, Vis a cozero set since (f(x)-δ,1]is open in [0,1], and Fis a zero set because [f(x)-δ,1]is closed in [0,1]. □
Theorem 4.3 (Baire Category–Type Theorem). Let Xbe a pseudocompact space, and let {U_n:n∈N}be a sequence of almost dense cozero sets in X. Then the intersection
⋂_(n=1)^∞ U_n
is a nonempty almost dense subset of X.
Proof. Let D=⋂_(n=1)^∞ U_n. We first show that D≠∅and then prove that it intersects every nonempty cozero set.
Take any nonempty cozero set V⊂Xand pick x∈V. Since U_1is almost dense, V∩U_1≠∅and is itself a cozero set. Choose x_1∈V∩U_1. By Theorem 4.2, there exist a cozero set V_1and a zero set F_1such that
x_1∈V_1⊆F_1⊂V∩U_1.
Continuing recursively, for each n≥1, select x_(n+1)∈V_n∩U_(n+1)and construct cozero sets V_(n+1)and zero sets F_(n+1)satisfying
x_(n+1)∈V_(n+1)⊆F_(n+1)⊂V_n∩U_(n+1).
The sequence {F_n}forms a nested collection of nonempty zero sets. By Theorem 4.1 and pseudocompactness of X,
⋂_(n=1)^∞ F_n≠∅.
Since each F_n⊆⋂_(k=1)^n U_k∩V, we conclude that D∩V≠∅. As Vwas an arbitrary nonempty cozero set, Dis almost dense. □
Remarks on Separability Properties
It is well-known that separability is not hereditary, as demonstrated by Niemytzky’s plane. Similarly, almost separability is not hereditary; the Niemytzky plane also provides a counterexample in this context.
We now summarize the relationships among various notions of separability. From standard results ([1, Section 1.2]):
"Strong sequentially separable "⟹" Sequentially separable "⟹" Separable".
From Theorem 3.2 in this paper, every separable space is almost separable. Combining these results, we obtain the chain of implications:
"Strong sequentially separable "⟹" Sequentially separable "⟹" Separable "⟹" Almost separable".
This establishes the hierarchical relationship among different separability concepts, clarifying the position of almost separable spaces in the broader topology framework.
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