This chapter looks at some ways in which nonstandard methods can be applied to problems in geometric topology. It considers some standard notions of connectedness which are similar to the classical “path-connected” and natural from the nonstandard point of view. The chapter provides an example of an approximation argument which is simplified by the existence of the infinitesimals. It looks at a well-known open question about sets which have the fixed point property, and aims to present some indication of why nonstandard methods may prove to be helpful in considering such a problem. The chapter also considers some notions of connectedness which are similar to path-connected and yet closer to capturing the notion of connectedness. The concept of “path-connected” is intuitively very simple, whereas the more general notion of “connected” is defined in a cumbersome way.