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_{1}spaces, Hausdorff spaces, etc.

I more or less get the formal definition, but I can't quite grasp the intuition behind them.

Any help and insights would be highly appreciated.

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- Thread starter beep300
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In summary, the countability and separation axioms in general topology are a way of making generalizations and simplifications when proving theorems. They allow mathematicians to ask whether certain properties are really necessary, and to make the least convenient assumptions when proving a theorem. These axioms also help define the applicability of a theorem. The first-countable and second-countable properties are distinctions between local and global properties, and they are useful in proving the existence of certain objects in a topological space. However, these properties do not always hold in all topological spaces, particularly in infinite-dimensional normed vector spaces.

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I more or less get the formal definition, but I can't quite grasp the intuition behind them.

Any help and insights would be highly appreciated.

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Since any union of open sets is open again but only a finite intersection is and vice versa for closed sets, and compactness is defined by the coverage with open sets, it's natural to speak of different countables, i.e. how many open sets it takes for something.

The same for separation properties. With a metric it's easy to see whether something is disjoint or not. Just measure it. But without a metric? What do we actually have to separate? Points, sets, which sets, points from sets and so on. Therefore the distinction between the various separation axioms is needed. It condenses theorems to a list of conditions which are really needed to proof a statement and therefore defines its applicability. When you deal with special topologies and all of a sudden you find out that you have only a ##T_1## space, you might get nervous and need to have a closer look on what you might take for granted and what not. Hausdorff is sometimes already a luxury.

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fresh_42 said:

Since any union of open sets is open again but only a finite intersection is and vice versa for closed sets, and compactness is defined by the coverage with open sets, it's natural to speak of different countables, i.e. how many open sets it takes for something.

The same for separation properties. With a metric it's easy to see whether something is disjoint or not. Just measure it. But without a metric? What do we actually have to separate? Points, sets, which sets, points from sets and so on. Therefore the distinction between the various separation axioms is needed. It condenses theorems to a list of conditions which are really needed to proof a statement and therefore defines its applicability. When you deal with special topologies and all of a sudden you find out that you have only a ##T_1## space, you might get nervous and need to have a closer look on what you might take for granted and what not. Hausdorff is sometimes already a luxury.

Thanks, I get it a lot better now.

There's one thing I don't get though: What's the distinction between first-countable and second-countable spaces? What's the intuition behind them?

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Again, in the real world, i.e. ##ℝ## or in ##ℝ-##vector spaces everything is fine already: both are valid.beep300 said:Thanks, I get it a lot better now.

There's one thing I don't get though: What's the distinction between first-countable and second-countable spaces? What's the intuition behind them?

If you take an uncountable set with the discrete topology, then it's not second countable. Another example can be found in the Sorgenfrey line.

Basically first countability is a local property (Every point has ...) and second countability a global property (The topological space has ...).

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The first countable property is basically an abstraction from metric spaces.beep300 said:Thanks, I get it a lot better now.

There's one thing I don't get though: What's the distinction between first-countable and second-countable spaces? What's the intuition behind them?

In any topological space, a point x is in the closure of a set A if there exists a sequence of elements of A converging to x. Also a function f between topological spaces preserves convergent sequences if it is continuous. In the presence of first countability both of the above become iff's.

Second countability is a good property to have because it can be used to prove that certain objects exist. In fact, regularity and second countability imply metrizability of a topological space, so here you have two of the properties in your list being used.

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fresh_42 said:Again, in the real world, i.e. ##ℝ## or in ##ℝ-##vector spaces everything is fine already: both are valid.

For finite-dimensional vector spaces that is. Many infinite-dimensional normed vector spaces are not second countable.

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Thanks everyone, I get the first and second countability axioms better now.

I guess one shoud always look at how definitions from general topology apply to intuitive topological spaces (such as the metric space) to get a better grasp.

Could one state this property, in other words: that given some subset A, x is in its closure iff some sequence in A converges to x?

This iff condition seems far more intuitive than the original definition I was provided with in the textbook.

I guess one shoud always look at how definitions from general topology apply to intuitive topological spaces (such as the metric space) to get a better grasp.

Cruz Martinez said:The first countable property is basically an abstraction from metric spaces.

In any topological space, a point x is in the closure of a set A if there exists a sequence of elements of A converging to x. Also a function f between topological spaces preserves convergent sequences if it is continuous. In the presence of first countability both of the above become iff's.

Second countability is a good property to have because it can be used to prove that certain objects exist. In fact, regularity and second countability imply metrizability of a topological space, so here you have two of the properties in your list being used.

Could one state this property, in other words: that given some subset A, x is in its closure iff some sequence in A converges to x?

This iff condition seems far more intuitive than the original definition I was provided with in the textbook.

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