| Popis: |
It is well know that the Tychonoff product of 2 ω many separable spaces is separable [2] , [3] . We consider for the Tychonoff product of 2 ω many separable spaces the problem of the existence of a dense countable subset, which contains no nontrivial convergent in the product sequences. The first result was proved by W.H. Priestley. He proved [14] that such dense set exists in the Tychonoff product ∏ α ∈ 2 ω I α of closed unit intervals. We prove ( Theorem 3.2 ) that such dense set exists in the Tychonoff product ∏ α ∈ 2 ω Z α of 2 ω many Hausdorff separable not single point spaces. We prove that in ∏ α ∈ 2 ω Z α there is a countable dense set Q ⊆ ∏ α ∈ 2 ω Z α such that for every countable subset S ⊆ Q a set π A ( S ) is dense in a face ∏ α ∈ A Z α for some A, | A | = ω . We prove ( Theorem 3.4 ) that in ∏ α ∈ 2 ω I α there is a countable set, that is dense but sequentially closed in ∏ α ∈ 2 ω I α with the Tychonoff topology and is closed and discrete in ∏ α ∈ 2 ω I α with the box topology ( Theorem 3.4 ). |
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