Several Spaces With Property (A)

In mathematics, metric space has been widely used. So, metrizable space is an im-portant question in general topology. During the study of metrizable space, (A) space, (G) space and (F) space are founded.They all have many good topological properties and have been studied by mathematicians.In this passage, there are four chapters. And the main idea is the property of some kinds of spaces with property (A).Chapter 1 is introduction. In this chapter the history of (A) space is showed and the definitions of (A) space, (G) space and (F) space are given. And some properties of (A) space and metrizable space can be found.In Chapter 2, we discuss decreasing (A) space, uniform (A) space, neighborhood (A) space, weak (A) space, decreasing weak (A) space, uniform weak (A) space and neighbor-hood weak (A) space. And get the equivalent condition of these spaces.The image of these space under some maps are given in Chapter 4. We see that the open and pseudo-open maps preserve weak (A) space. But they cannot preserve (A) space. Closed finite-to one maps can preserve decreasing weak (A) space.In the end we obtain a property in decreasing decreasingα-(F) space.