EMBEDDINGS OF THE T_k-TOPOLOGICAL SPACES

다국어 초록 (Multilingual Abstract)

Let (ℤ, T_k) be a topological space on the set of integers, where the topology T_k is generated by the set S_k as a subbase, where k ∈ ℤ and S_k := {S_k,t | S_k,t := {2t, 2t + 1, 2t + 2k + 1}, t ∈ ℤ}. Then we call (ℤ, T_k) a T_k-topological space. For the set of positive integers k₁, k₂ ∈ ℕ, the paper initially proves that (ℤ, T_k₁) is topologically embedded into (ℤ, T_k₂) if and only if k₁ ≤ k₂, i.e., ∃ an embedding h_(k₁,k₂) : (ℤ, T_k₁) → (ℤ, T_k₂) ⇔ k₁ ≤ k₂. Furthermore, in the case of k₁ ≤ k₂, the subspace (ℤ \ Im(h_(k₁,k₂)), (T_k₂)_{ℤ\Im(h_(k₁,k₂))}) induced from (ℤ, T_k₂) has (k₂ - k₁) components. Besides, each of the components is homeomorphic with the Khalimsky (K-, for brevity) topological line. Finally, we obtain some embeddings of (ℤⁿ, κⁿ) into (ℤⁿ, (T_k)ⁿ), the n-fold product topological space of a T_k-topological space, k ∈ ℕ, where (ℤⁿ, κⁿ) is the n-dimensional K-topological space. Since a T_k-topological space is an Alexandroff space, an embedding problem of a T_k-topological space plays an important role in pure and applied topology.