On strict and simple type extensions
Let (Y,τ) be an extension of a space (X,τ′)⋅p∈Y, let 𝒪yp={W∩X:W∈τ,p∈W}. For U∈τ′, let o(U)={P∈Y:U∈𝒪yp}. In 1964, Banaschweski introduced the strict extension Y#, and the simple extension Y+ of X (induced by (Y,τ)) having base {o(U):U∈τ′} and {U∪{p}:p∈Y,and U∈Oyp}, respectively. The extensions Y# and...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
1998-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1155/S0161171298000349 |
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| Summary: | Let (Y,τ) be an extension of a space (X,τ′)⋅p∈Y, let 𝒪yp={W∩X:W∈τ,p∈W}. For U∈τ′, let o(U)={P∈Y:U∈𝒪yp}. In 1964, Banaschweski introduced the strict extension Y#, and
the simple extension Y+ of X (induced by (Y,τ)) having base {o(U):U∈τ′} and
{U∪{p}:p∈Y,and U∈Oyp}, respectively. The extensions Y# and Y+ have been extensively used since
then. In this paper, the open filters
ℒp={W∈τ′:W⫆intxclx(U) for some U∈𝒪yp}, and 𝒰p={W∈τ′:intxclx(W)∈𝒪yp}={W∈τ′:intxclx(W)∈ℒp}=∩{𝒰:𝒰 is an open ultrafilter on X,𝒪yp⊂𝒰}
on X
are used to define some new topologies on Y. Some of these topologies produce nice
extensions of (X,τ′). We study some interrelationships of these extensions with Y#, and Y+ respectively. |
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| ISSN: | 0161-1712 1687-0425 |