Embedding TopSys into a topological category

The purpose of this paper is to make a case for the value of many-valued mathematics, often called fuzzy mathematics. We believe there may be a difference between many-valued mathematics and fuzziness, as used by those who work with fuzzy logic and fuzzy set theory and applications thereof. We think that most, if not all, fuzzy mathematics is many-valued. However, for this paper, the difference between many-valued mathematics and fuzzy mathematics, if a difference exists, is not important. We are, in this paper, content to show that many-valued mathematics can contribute to mathematics. We do understand that for those mathematicians who feel that many-valued mathematics does not have a place in mathematics this paper will not cause them to embrace many-valued mathematics, but we would like them to consider that many-valued mathematics might be able to contribute to mathematics. In this paper, we give an example of a mathematical construction which was created and defined in part to help computer scientists understand and be able to use topological ideas and concepts in their work as computer scientists. Thus, one would think that this construction, called topological systems, would be topological (as defined later). However, it seems that topological systems are clearly not topological. Thus, an interesting question is can topological systems be made topological, or said more mathematically, can topological systems be embedded into something which is topological. We answer this question in the affirmative, and we do it by embedding topological systems into something which is many-valued. It may be the case that someone(s) can some day show that topological systems are topological though this seems unlikely. Or it may be the case that someone(s) can embed topological systems into something which is topological but not many-valued. However, our point is that by using something which is many-valued we have added to mathematics, and thus, we have shown a mathematical use of many-valued mathematics. We should also say that the mathematical results in this paper are not new. We do present some ideas, including the motivation for the “topological” embeddings from topological systems, in new, and we think, illuminating ways, but the mathematical results are not new.