Extending maps defined on dense sets

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In summary, to extend a map from a dense space into another dense space, you need to find a function that is uniformly continuous and has a Cauchy sequence as its image.

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Say you have a (continuous) map from a space A into a space Y, where A is dense in some other space X. When is it possible to extend this to a map from X into Y? Intuitively, to find the value of the map at a limit point, just take the limit of the values on a sequence approaching it. However, it's not clear the resulting sequence will always have a limit (eg, 1/x or sin(1/x) on (0,1), which is dense in [0,1)), and even when it does, is the resulting function always continuous?

The resulting function might not even be well-defined. A=R\{0} is dense in X=R. Consider the map from A to [0,1] which sends positive numbers to 1 and negative numbers to 0. There are sequences in A which converge to 0 that are strictly positive (hence the sequence of their values converges to 1), strictly negative (so the sequence of values converges to 0) and alternating between negative and positive (so the sequence of values doesn't converge).

But suppose that for all x in X\A there exists a unique y in Y such that if (an) is any sequence in A converging to x, (f(an)) converges to y. Then you can extend f to say f(x) = y. Then it shouldn't be hard to prove that this extend f is continuous, I think.

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First of all, what kind of space are we working in? For X and Y metric spaces, here's what I'm thinking:

If the function f is uniformly continuous, it sends Cauchy sequences to Cauchy sequences, so if we select a point x in X and find a sequence in A converging to x, that sequence is Cauchy so its image is Cauchy. If we further demand Y is complete, then the image sequence converges, to some y in Y. If F is the extension of f to X, let y be F(x). I don't see anything going wrong with the verification that this is well defined and continuous on all of X. Thus f uniformly continuous and Y complete ensures the extension exists. These may not be the weakest conditions that ensure this happens though. Also note that if X is compact, then f is automatically uniformly continuous.

1. What is the definition of a "dense set"?

A dense set is a subset of a topological space where every point in the space is either a limit point or an isolated point of the subset. In other words, every point in the space is either in the subset or arbitrarily close to it.

2. Why is it important to extend maps defined on dense sets?

Extending maps defined on dense sets allows us to define the map on the entire topological space, even if the original map was only defined on a subset. This can be useful in many mathematical applications, such as in analysis and topology.

3. What are some common techniques for extending maps defined on dense sets?

One common technique is the use of continuity, where we extend the map by requiring it to be continuous on the entire space. Another technique is the use of completeness, where we extend the map by requiring it to preserve certain properties, such as Cauchy sequences.

4. Can a map defined on a dense set always be extended to the entire space?

No, not all maps defined on dense sets can be extended to the entire space. The ability to extend a map depends on the properties of the map and the space it is defined on. In some cases, it may not be possible to extend a map in a meaningful way.

5. How does extending maps defined on dense sets relate to the concept of continuity?

Extending maps defined on dense sets is closely related to continuity, as mentioned in question 3. Continuity is often a key factor in being able to extend a map, as it allows for the map to be defined on the entire space in a consistent and meaningful way.