Abstract: In this article, the researcher shows a survey on Paracompact open top for function sps such as the Paracompact open top for function sp, sequential properties of function sp with the Paracompact open top, and locally Paracompactness in function sps with the Paracompact open topology is studied. Some applications of Paracompact open top for function sps and its properties are also provided. Also, the researcher studies the Paracompact open top for the set K^X of all functions which are continuous from X to K and the special case of all continuous real valued functions on a Tychonoff sp X. The researcher compares this top with several the well-known tops. Also, the researcher introduces the product topology in function sps. Finally, on the Paracompact open and admissible tops, the Paracompact open top and loop sps are studied, where topological spaces is denoted by top sp.

Keywords: Top space, Paracompact sp., Sequential, Locally Paracompactness, Admissible, Loop sps, Function sps, Lindelof sps. Continuous, Real valued function, Compact open top

Pages: 538-541