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pellman
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What is getting separated from what? I presume there is some historical founding case that involved separating something. Like how the original vector spaces were mental arrows in R^3.
micromass said:So the idea is to approximate certain elements by other certain elements. Separable means that any element of the set can be approximated by a limited number of elements...
micromass said:So the idea is to approximate certain elements by other certain elements. Separable means that any element of the set can be approximated by a limited number of elements.
For example, with ##\mathbb{R}##, we can approximate an arbitrary element (for example ##e##) by their decimal representation. So we can approximate ##e## better and better by
[tex]2. ~2.7, ~ 2.71, ~ 2.718, ...[/tex]
How do you use that analogy in non-metrizable separable spaces?
We can do exactly the same with any real number. We can do this in two dimensions too. For example, the couple ##(e,\pi)## can be approximated by
[tex](2,3),~(2.7,3.1),~(2.71, 314),...[/tex]
So separability actually is a very far-reaching generalization of the decimal representation. So basically, we have a countable number of "basis" elements, and then we succeed in approximating any element by the basis elements.
In the theory of Hilbert spaces, the analogy becomes even better with the existence of countably orthonormal bases. But I can only tell this if you know Hilbert spaces.
In a very general sense, you should see separable spaces and spaces which are "not too large" and where countable many terms suffice in many cases. In the same way that rational numbers can be used to describe real numbers.
It turns out that a very large portion of all spaces encountered "in nature" are separable. This is especially true in physics. This is good because it allows us to describe a physical system by countably many terms (like position, momentum, etc.) instead of dealing with uncountabilities.
WWGD said:Well sure, but I don't see how the analogy is helpful in non-metrizable cases.
A space is considered separable if it contains a countable, dense subset. This means that there exists a subset of the space that is countably infinite and its closure is the entire space.
Separable spaces play a critical role in many areas of mathematics, including analysis, topology, and functional analysis. They allow for the construction of important mathematical objects such as countable bases and dense subspaces, and they also have many useful properties that make them easier to study and work with.
Some separable spaces can also be compact, meaning they are both countably infinite and have a finite number of elements. However, not all separable spaces are compact, and not all compact spaces are separable. Whether a space is separable or compact depends on its specific properties and topological structure.
No, not all metric spaces are separable. For a metric space to be separable, it must contain a countable, dense subset. If the space does not have such a subset, it cannot be considered separable. However, many commonly studied metric spaces, such as Euclidean spaces and finite-dimensional vector spaces, are separable.
The term "separable" comes from the fact that these spaces have a countable, dense subset, which can be thought of as "separating" or "dividing" the space into smaller, more manageable parts. This property also helps to distinguish these spaces from others that may not have a countable, dense subset.