We didn't get to the punchline of Friday's lecture, so I'm not sure what I am supposed to be thinking about the definitions we created.

One thing I notice about definitions is that some are illustrative and some are concrete. What I mean by that is, when we define a cauchy set, we really need that concrete definition forever so that we can always be on the same page about this property. However, when we define our d, we are just giving one illustration of what d can be, and d can be different things at different times... we are just stating what it is today. The problem with this is that there is not always a clear distinction when we are making a definition what action we should take with it... whether it is permanent and we should commit it to memory, or transitory - and just use it for now to help us think about the topic at hand.

This is not the case for Friday's definitions. Clearly we should memorize (or have already memorized) the definitions for:

• when a mapping and its transform are continuous
• what a similarity transform is
• what is required of a set in order for it to be called self-similar
• when a sequence is cauchy
• when a space is complete
• when a space is compact
• what one means by "distance from a point to a set"
• what one means by "jointly continuous"
• what one means by "distance between two sets"

Are the words space and set exchangeable in R^2?

When we talk about a sequence being a subset of a space, - such as "the sequence x_n is an element of X, where X is a metric space," are we implying that X = R^2 - or can X be anything and contain sequences? - in other words, are sequences abstract or are sequences only defined by numbers.