One joy in speaking to the Mathematics Section of the New York Academy of Science5 is the opportunity to present an illustrated talk to an audience that appreciates mathematical content and demands liveliness. I am happy to say that I'm not alone in noticing the possibilities of the proper name in the title.(For example, a barefy Bairr space X is a Baire space such that its product X x Y with some Baire space Y is not a Baire space [5].) I illustrated my talk accordingly. Unfortunatcly, the lively illustrations have been lost, and the reader must supply them from a fcrtile imagination.

My main intention in this paper is to present a mathematical clan consisting of two families of examples of the fruitful and attractive blending of topology and vector spaces. The first family has been studied at various times by Steve Saxon and myself both independently and jointly. I gratefully acknowledge my introduction to this work by Steve while he was my advisor and his constant encouragement since then. The second family has looser requirements than the first and has been carefully studied by W. Lehner [ll] in their application to spaces of continuous functions. I shall introduce the concepts from linear spaces and from general topology that are central to understanding this work, and I will largely restrict myself to statements that arc understandable in this context. Examples, observations, and theorems, but few proofs, will find their way into this discussion. This is intentional, as references will provide the needed proofs or the techniques for constructing them, and I wish to provide an overview of this area of research. Thc basic information about linear topological spaces is admirably described in the following works: Horvath [8], Kelley el al.[la], and Robertson and Robertson [161.