Some Results Related with Semi Compactness in Bitopological Spaces

Abstract

The triple (𝑋, 𝜏1, 𝜏2) is known as a bitopological space, where 𝜏1 and 𝜏2 are two topologies which are defined in a nonempty set 𝑋. The concept ‘bitopological space’ was established from asymmetric metric spaces. The objective of this paper is to establish some results which are related with 𝛿 −semi compactness in bitopological spaces. In particular, we can identify the relationship between the bitopological spaces and their product space in semi compactness. For a pairwise 𝛿 −continuous surjective and pairwise 𝛿 − open mapping 𝑓 ∶ (𝑋, 𝜏1, 𝜏2) ⟶ (𝑌, 𝜎1, 𝜎2), the image of a 𝜏1𝜏2 − 𝛿 semi compact space under 𝑓 is 𝜎1𝜎2 − 𝛿 semi compact space. Furthermore, the product space (𝑋 × 𝑌, 𝜏1 × 𝜎1, 𝜏2 × 𝜎2) is 𝜏1 × 𝜎1𝜏2 × 𝜎2 − 𝛿 semi compact space, if both (𝑋, 𝜏1, 𝜏2) and (𝑌, 𝜎1, 𝜎2) are 𝜏1𝜏2 − 𝛿 semi compact and 𝜎1𝜎2 − 𝛿 semi compact respectively. Moreover, if a bitopological space (𝑋, 𝜏1, 𝜏2) is 𝜏1𝜏2 − 𝛿 semi compact and topological spaces (𝑋, 𝜏1) and (𝑋, 𝜏2) are 𝛿 −Hausdorff space then the semi regularization of 𝜏1 and 𝜏2 are equal. That is, 𝜏1𝑠 = 𝜏2𝑠 . Through these results, we are able to get the clear understanding about the concept ‘semi compactness’ and how to connect this concept with topological spaces and bitopological space. In addition, we can identify the way to connect the continuous maps and product spaces with semi compactness.