The general fuzzy filter approach to fuzzy topology, I

Initial and Final Characterized Fuzzy 1 3 2 T and Finer Characterized Fuzzy 12 2 R -Spaces

“…Recall that a pre filter on X [17] is a non-empty proper subset of  of L X such that (1) µ, η ∈  X implies µ ∧ η ∈  and (2) from µ ∈  and µ ≤ η it follows η ∈ .  subset  of L X is said to be superior fuzzy filter base [2] if the following conditions are fulfilled:…”

“…A family (B α ) α ∈ L0 of non-empty subsets of L X is called a valued fuzzy filter base [2] if the following conditions are fulfilled:…”

“…[2], each valued fuzzy filter base ( α ) α ∈ L defines an fuzzy filter  on X by = ∨ for all µ ∈ L X . Conversely, each fuzzy filter  can be generated by a valued fuzzy filter base, e.g., by (α-pr ) α ∈ L0 with α-pr M={µ∈L X ⃒ α≤(µ)}.…”

“…Suprema and infimum of sets of fuzzy filters are meant with respect to the finer relation. An fuzzy filter  on X is said to be ultra [2] fuzzy filter if it does not have a properly finer fuzzy filter. For each fuzzy filter  ∈  L X there exists a finer ultra fuzzy filter U ∈  L X such that U ̸ ≤ .…”

“…The fuzzy filter on X [14] is the mapping M: L X →L such that the following conditions are fulfilled: (F1) ( ) α α Μ ≤ for all α ∈ L and (1)=1.…”