General topology: Countability and separation axioms

[beep300]

I need some help understanding the countability and separation axioms in general topology, and how they give rise to first-countable and second-countable spaces, T₁ 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.

 

[fresh_42]

I'm not sure if I have a satisfactory answer for you. I have always thought of this as simply a form of generalizations. The world we live in is a metric world and many things are covered by school math. We measure distances, open sets are simply spaces like intervals without the boundary, all is somehow dense and uncountable and so on. Not that there aren't enough weird topological facts in it despite of that, e.g. Cantor sets. However, if we start to state a topological theorem, it is for mathematicians somehow self evident to ask whether all this metric induced properties are really needed. Mathematicians love counterexamples and exceptions. So it is only natural to make the least instead of the most convenient assumptions when proving a theorem - and it's expanding its usage!

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.

 

[beep300]

I'm not sure if I have a satisfactory answer for you. I have always thought of this as simply a form of generalizations. The world we live in is a metric world and many things are covered by school math. We measure distances, open sets are simply spaces like intervals without the boundary, all is somehow dense and uncountable and so on. Not that there aren't enough weird topological facts in it despite of that, e.g. Cantor sets. However, if we start to state a topological theorem, it is for mathematicians somehow self evident to ask whether all this metric induced properties are really needed. Mathematicians love counterexamples and exceptions. So it is only natural to make the least instead of the most convenient assumptions when proving a theorem - and it's expanding its usage!

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?

 

[fresh_42]

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?

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

 

[Cruz Martinez]

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?

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.

 

[micromass]

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.

 

[beep300]

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.

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.