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so the first and most fundamental thing that were going to need in order to talk about how close two real numbers are or two things are with an analysis and topology is a notion of interior point what is it going to mean for a point to be an interior point of a set this is going to be an important notion because its going to let us speak of which points inside of a set have a little bit of breathing room around them and which ones do not its those points with a little bit of breathing room that we call the interior points and these are going to be fundamental to our definition of what gets to be an open set lets take a look at how it works so our story begins with a subset a of the real numbers and we dont know what the subset looks like maybe its kind of nasty im just going to make it kind of this purple cloud of real numbers over here as well as a point x that is inside of that subset the question is what do we need in order to come up with a reasonable notion of what it mean
