Infinite Product Spaces Under the Tychonoff and Goofynoff Topologies
The only topology considered for the infinite product of topological spaces in most current topology texts and research papers is the Tychonoff topology. Yet there is another topology which seems to be a much more topologically natural generalization of the usual "box" topology of finite products. We call this natural generalization the Goofynoff topology and exploit its properties. The use of the word "Goofynoff" (pronounced Goof'-n-off) is not universal and does not refer to any person of
... name. In the few references to this topology that can be found, it is usually called simply the Box Topology. None of the popular topology texts provide much indication of why the Tychonoff topology is generally chosen instead of the more natural Goofynoff topology. This paper provides the reasons from a topologist's point of view. In Section III of this paper we compare the productivity of various properties under these two topologies. This paper was motivated by a question asked in the notes for a topology class at Utah State University in the fall of 1972. This question, concerned with the connectedness of an infinite product of connected spaces (under the Goofynoff topology), went unanswered through two graduate classes. An example showing that such a space may fail to be connected is given in Section II. In some respects the example resembles the Hilbert Cube Iw and, for that reason, is called the Goofy Cube and is denoted by Gw. In order to emphasize the difference, we include a table at the end of Section II which compares the topological properties of Iw with those of Gw.