  In this paper, we introduce the concept of quasi-point-separable topological
vector spaces, which has the following important properties: 1.In general, the
conditions for a topological vector space to be quasi-point-separable is not
very difficult to check; 2.The class of quasi-point-separable topological
vector spaces is very large that includes locally convex topological vector
spaces and pseudonorm adjoint topological vector spaces as special cases;
3.Every quasi-point-separable Housdorrf topological vector space has the fixed
point property (that is, every continuous self-mapping on any given nonempty
closed and convex subset has a fixed point), which is the result of the main
theorem of this paper (Theorem 4.1). Furthermore, we provide some concrete
examples of quasi-point-separable topological vector spaces, which are not
locally convex. It follows that the main theorem of this paper is a proper
extension of Tychonoffs fixed point theorem on locally convex topological
vector spaces.
