We consider topological properties based on continuous extensions of total preorders. First we study extension properties based on closed subsets of topological spaces. Thus, we analyze the Yi’s extension property for which any total preorder defined on a closed subset of a topological space, and continuous with respect to the relative topology, admits a continuous extension to the whole space. We show that the Yi’s extension property implies the Tietze’s extension property, so that it is a particular case of normality. In the second part of this paper, we study extension properties based on open subsets of topological spaces, showing that they are related to topological properties of extremal disconnectedness.
We consider the problem of extending preferences from a subset of a commodity space to the entire space. It is a simple consequence of the Tietze extension theorem that continuous preferences can be extended if they are defined on closed subsets of a normal space and are representable by utility functions. We show the following: If the space is a non-separable metric space, then extension of preferences is not always possible. In fact for (path-connected) metric spaces, extension property, utility representation property, and separability are equivalent to each other.
We consider the problem of extending preferences from a subset of a commodity space to the entire space. It is a simple consequence of the Tietze extension theorem that continuous preferences can be extended if they are defined on closed subsets of a normal space and are representable by utility functions. We show the following: If the space is a non-separable metric space, then extension of preferences is not always possible. In fact for (path-connected) metric spaces, extension property, utility representation property, and separability are equivalent to each other.