Semi-closure Spaces

Book excerpt: The first chapter of this thesis expands on the semi-closure operator discussed by Hong in his paper, and in particular how sets behave in a semi-closure space. Hong notes that a semi-closure space is a generalized topological space, and in this thesis, an example of a semi-closure space is shown to not be topological; i.e. semi-closure spaces are less restrictive. In the second chapter, continuity of functions between two semi-closure spaces, separation of points within a semi-closure space, and compactness of semi-closure spaces are established. An example of a T2 semi-closure space is found to not be topological, again showing semi-closure spaces to less restrictive. The third chapter explores how extensions of topologies discussed by Hicks can be applied to semi-closure spaces. By intersecting closed subsets from semi-closure subspaces with arbitrary sets in the larger space, we increase, or extend, the number of closed sets in our space.