This paper describes the definition and testing of a new type of median filter for images. The topological median filter implements some existing ideas and some new ideas on fuzzy connectedness to improve, over a conventional median filter, the extraction of edges in noise. The concept of alpha-connectivity is defined and used to create an algorithm for computing the degree of connectedness of a pixel to all the other pixels in an arbitrary neighborhood. The resulting connectivity map of the neighborhood effectively disconnects peaks in the neighborhood that are separated from the center pixel by a valley in the brightness topology. The median of the connectivity map is an estimate of the median of the peak or plateau to which the center pixel belongs. Unlike the conventional median filter, the topological median is relatively unaffected by disconnected features in the neighborhood of the center pixel. Four topological median filters are defined. Qualitative and statistical analyses of the four filters are presented. It is demonstrated that edge detection can be more accurate on topologically median filtered images than on conventionally median filtered images.