On Variations of Fuzzy Homogeneity

Some of the important concepts in ordinary topological spaces like <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>n</mml:mi> </mml:math> -homogeneity, strongly locally homogeneity, almost countable dense homogeneity, and transposition homogeneity are extended to fuzzy topological spaces and the relationship between these spaces together with fuzzy semi-discreteness is well studied here. Since there are spaces that fail to be fuzzy countable dense homogeneous ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>F</mml:mi> <mml:mi>C</mml:mi> <mml:mi>D</mml:mi> <mml:mi>H</mml:mi> </mml:math> ) or fuzzy semi discrete ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>F</mml:mi> <mml:mi>S</mml:mi> <mml:mi>D</mml:mi> </mml:math> ), only due to finitely many points, we found it necessary to study such spaces, which we refer to as almost fuzzy countable dense homogeneous ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>A</mml:mi> <mml:mi>F</mml:mi> <mml:mi>C</mml:mi> <mml:mi>D</mml:mi> <mml:mi>H</mml:mi> </mml:math> ) and almost fuzzy semi discrete spaces ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>A</mml:mi> <mml:mi>F</mml:mi> <mml:mi>S</mml:mi> <mml:mi>D</mml:mi> </mml:math> ) respectively. It is shown that the fuzzy extensions of the above notions are all fuzzy topological properties. A further variant of almost countable dense homogeneity called <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>A</mml:mi> <mml:mi>F</mml:mi> <mml:mi>C</mml:mi> <mml:mi>D</mml:mi> <mml:msup> <mml:mi>H</mml:mi> <mml:mo>*</mml:mo> </mml:msup> </mml:math> was introduced to fuzzy spaces and proved that a countable fuzzy separable space is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>A</mml:mi> <mml:mi>F</mml:mi> <mml:mi>S</mml:mi> <mml:mi>D</mml:mi> </mml:math> if and only if it is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>A</mml:mi> <mml:mi>F</mml:mi> <mml:mi>C</mml:mi> <mml:mi>D</mml:mi> <mml:mi>H</mml:mi> </mml:math> or <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>A</mml:mi> <mml:mi>F</mml:mi> <mml:mi>C</mml:mi> <mml:mi>D</mml:mi> <mml:msup> <mml:mi>H</mml:mi> <mml:mo>*</mml:mo> </mml:msup> </mml:math> . Hence, in countable fuzzy separable spaces, the notions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>A</mml:mi> <mml:mi>F</mml:mi> <mml:mi>C</mml:mi> <mml:mi>D</mml:mi> <mml:mi>H</mml:mi> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>A</mml:mi> <mml:mi>F</mml:mi> <mml:mi>C</mml:mi> <mml:mi>D</mml:mi> <mml:msup> <mml:mi>H</mml:mi> <mml:mo>*</mml:mo> </mml:msup> </mml:math> are equivalent. One of the main results identifies conditions under which an <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>A</mml:mi> <mml:mi>F</mml:mi> <mml:mi>C</mml:mi> <mml:mi>D</mml:mi> <mml:mi>H</mml:mi> </mml:math> space becomes an <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>F</mml:mi> <mml:mi>C</mml:mi> <mml:mi>D</mml:mi> <mml:mi>H</mml:mi> </mml:math> space. Further, the paper discusses ordinary topological spaces generated by fuzzy topological spaces and we shall observe that whenever the fuzzy topological space is almost fuzzy semi discrete, or fuzzy transposition homogeneous or almost fuzzy countable dense homogeneous, so is the corresponding ordinary space.