Separation Axioms of Center Topological Space

Quotient and Product of Center Topological Groups

“…Let (𝑋, 𝛿) be a proximity space and {𝑥}, 𝐵 ⊆ 𝑋, such that {𝑥}𝛿𝐵. Then 𝑥 𝐵 = {〈{𝑥}, 𝐵〉} is called a center point in 𝑋 [6]. Definition 1.7 [6] Let 𝑥 𝐵 be a center point in (𝑋, 𝛿) and 𝒞 𝐴 center set in (𝑋, 𝛿).…”

“…Then 𝑥 𝐵 = {〈{𝑥}, 𝐵〉} is called a center point in 𝑋 [6]. Definition 1.7 [6] Let 𝑥 𝐵 be a center point in (𝑋, 𝛿) and 𝒞 𝐴 center set in (𝑋, 𝛿). Then 𝑥 𝐵 ∈ 𝒞 𝐴 if and only if 〈𝐴, 𝐵〉 ∈ 𝒞 𝐴 and 𝑥 ∈ 𝐴 [6].…”

“…The triplet (𝑿, 𝜹 𝑿 , 𝝉 𝒄𝒆𝒏𝒕(𝑿) ) is called a center topological space and the members of 𝝉 𝒄𝒆𝒏𝒕(𝑿) are said to be 𝓒open. 𝝉 𝒄𝒆𝒏𝒕(𝑿) is called indiscrete center topology if 𝝉 𝒄𝒆𝒏𝒕(𝑿) = { 𝓒 𝑿 , 𝓒 ∅ } and called discrete 𝐜𝐞𝐧𝐭𝐞𝐫 toplogy if 𝝉 𝒄𝒆𝒏𝒕(𝑿) = 𝒄𝒆𝒏𝒕(𝑿) [6].…”

“…Cyclic compression and the optimal proximity point are two of the key concepts in the fixed point hypothesis, and various findings, like the reference [5], have been made. Kandil et al [1] most recently developed a new method to proximity structures that is based on the ideal and soft set ideas [6]. By utilizing proximity space, the center set was first introduced in reference [7].…”