Function Space And Its Generalized Metric Properties

The idea of topologizing the set of continuous functions from topological space into another topological space came from the notions of pointwise and uniform convergence of sequence of functions. The works of Ascoli [1883], Arzela [1889] and Hadamard [1898] marked the beginning of function space theory. In 1935, Tychonorff proved that the pointwise convergence topology is exactly the Tychonorff topology in R^X. Soon after that, Fox defined the compact-open topology, a generalization of pointwise convergence topology. Then, more and more topologists began the systematic researching of the function spaces, mainly in the pointwise convergence topology, compact-open convergence topology, and uniform convergence topology.In this paper, first, we investigate the generalized metric properties of the topological functional space. Ceder's famous paper "Some generalizations of metric spaces" was published in 1961, many topologists have devoted their attention to the question on M_i-space since then. In 1970's, Gruenhage[1976] and Junnila[1978] prove that M₃-spaces are M₂-spaces independently using different methods. Their result inspirits more enthusiasm of people, who want to solve the problem whether M₃-spaces are M₁-spaces.Recently, Paul M. Gartside and E.A. Reznichenko proved that C_k(P) is M₃ space where P is the irrational space. However, in the proof of their paper, there is no clue that C_k(P) is M₁ space, even when P is σ-compact and Polish space. Soon after that, G. Gruehage proved that C_k(X) is M₁ space if X is σ-compact Polish space and gave the construction of σ-closure preserving basis. In this...