### Continuity Introduced

### Topologies on a set and the maps preserving them

We've talked about the fact that in topology we classify spaces, where two spaces are equivalent if we can get from one to the other by stretching and squeezing. However, the notion of stretching and squeezing is of course no basis for a rigorous mathematical theory, and therefore we need to formalize this notion. It is convenient to think about what this means for any two topological spaces, and not only equivalent ones. This means that we essentially ask what kind of mappings we can have between any two topological spaces, which respect the topological structure (the same way that we want linear maps to preserve vector space structure). These are the continous maps. Forgetting the formal definition for now, the essence of this is the fact that we want maps that send points which are near in the domain to points which are near in the range.

### A Concrete Example - Two Maps on the Real Line

^{2}is clearly continuous.

It is also easy to give an example of a non-continuous function. We define a function g(x) which is zero everywhere except at the point 0, where we set g(0)=1. It is impossible to draw a graph for g(x) without lifting our pen from the paper at 0.

Let's try to understand the difference between the two cases, and from it try to get a definition of continuity. For f(x)=x

^{2}, every two points close to each other on the real line, are sent to two points close to each other on the graph. However, for g(x) this is no longer true, since the point 0 is sent to a point which is at a distance of 1 at least from the points close to it.

This definition is not that good for the real line, or for almost any space, because of the imprecision in defining what we mean by two points close to each other. Since for any two real numbers we have infinitely many real numbers between them, it makes no sense to talk about the number next to a number p. Instead, we establish a relationship between points on the line and sets of points on the line.

We now need some notations. An

**open interval**on the line, denoted by (a,b) for two real numbers a,b such that a < b, is the set of all points x such that

a < x < b. Given a point p on the line, a

**neighborhood**of p is an interval (a,b) such that a < p < b.

We can finally define nearness. We say that a point p is

**near**to a set A if every neighborhood of p has a point in common with A. For example, we can look at the point 0 and the point (0,1). 0 is near to the interval (0,1), since every neighborhood of 0 is of the form (a,b) for a < 0,b > 0 and this sort of neighborhood has many points in common with (0,1) (to be precise, the entire set of points (0,Min(1,b))).

Having defined near, we can finally talk about

**continuous**functions in a more formal manner. How do we turn the intuitive definition into a mathematically rigorous one? One way to do it is to say that a function f is continuous if for any point p and set A, f(p) is near to the set f(A)={f(a)a Î A}.

Using this definition, let's look at g(x) again. We now see that this function clearly violates the definition. Previously, we saw that 0 was near to (0,1) on the real line. However g(0)=1 but g(x)=0 for all x ¹ 0. This means that we need to ask if g(0)=1 is near to the set g((0,1))={0}, and it's clear that, for example, the neighborhood (1/2,2) of g(0)=1 has no point in common with (0,1).

To show that f(x)=x

^{2}is continuous takes some more work, but simple techniques from Calculus 1 suffice.

Those familiar with the definition of continuity for real functions might ask where has the dreaded e,d definition have run off to. The answer is simple. They are the same definition in the case of the real line. The advantage of the definition I'm using here is the fact that it generalizes to spaces where there is no notion of distance, and thus looking at all points close to a point has no meaning.

We can now look at continuity in the case of the real line, and see what's the only intrinsic information coming from the real line that we used in the definition. The point appearing in the definition is arbitrary, and so is the set we are checking nearness to. The only thing that we took as a given was the fact that a neighborhood of a point is an open interval. In the next posts we look at the abstract definition of neighborhood, and see that this completely defines the topology of the space.

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On 16 Aug 2006, 12:43.