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This research presents the concepts of compatibility and edge spaces in α-topological spaces, and introduces the α-topology combinatorially induced by the α-topology. Furthermore, studies the relationship between the α-topology on V ∪ E and the relative α-topology on V.

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This paper presents the concepts of prepaths, paths, and cycles in α-topological spaces and studies them in orderable spaces. Also, many relationships are proved with their equivalences using some properties in topological spaces like compactness and locally connectedness.

We discovered the embedding of K 6 in the Klein bottle is differed by the use of the link diagram, where the number of the components of the link diagram are not always equal. This result helps to construct some subfamilies of the embedding of Petersen family in the Klein bottle.