Examples for seperation axioms.

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In summary, the examples provided for the given assignments are as follows: 1. X=R, A=R-Q. 2. X=N, Y=N-{0}U{sqrt2}, and f:X->Y, f(x)=x if x in N-{0} and f(x)=sqrt2 if x=0. 3. X=Y with two different topologies, one finer than the other, and f=identity. These examples satisfy the given properties of Sep, S1, and S2.
  • #1
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My assignment is like this:
1.give an example of a space X and a subspace A of X s.t X satisifes Sep and A doesnt.
2.give an example of a continuous and onto function f:X->Y s.t X satisifies S1 but Y doesnt.
3.give an example of a continuous and onto function f:X->Y s.t X satisfies S2 and Y doesnt.

my answers are as follows:
1. X=R A=R-Q, is it a good example?
2.X=N, Y=N-{0}U{sqrt2} and f:X->Y f(x)=x if x in N-{0} and f(x)=sqrt2 if x=0, i think we can't find a countable set of bases for Y, not sure though.
3. didn't do it so far, any hints?
 
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  • #2
First, what do you mean by "separation"? I would consider all of the "Tychonoff" properties to be "separation" properties: T0: each singleton set is closed; T1: given any two points, there exist a set that contains one of them but not the other; T2 (Hausdorf): Given any two points there exist two disjoint open set such that one contains one point and the other set contains the other point.
 
  • #3
Sep says there's a countable dense set.
S1 says there's a countable basis at a point.
S2 says there's a countable basis for the topology.

those should be the countability axioms, sorry for misleading.
 
  • #4
"1." isn't good. The irrationals are actually a separable subspace of R (in the usual topology). In fact, if X is a separable metric space, then any subspace of X is separable as well. So your example is going to have to come from a non-metrizable topology.

For 2, what topology are you giving Y?

For 3, I would think about using X=Y, but giving it two topologies, one finer than the other. Then maybe using f=identity.
 

Related to Examples for seperation axioms.

1. What are the different types of separation axioms?

There are five main types of separation axioms: $T_0$, $T_1$, $T_2$, $T_3$, and $T_4$. These are also known as Kolmogorov axioms, and each one builds upon the previous one in terms of strictness.

2. What is the difference between $T_0$ and $T_1$ separation axioms?

The $T_0$ separation axiom states that for any two distinct points in a topological space, there exists at least one open set that contains one point but not the other. The $T_1$ separation axiom is stricter and requires that for any two distinct points, there exist two disjoint open sets, each containing one of the points.

3. Can you give an example of a space that satisfies the $T_0$ but not the $T_1$ separation axiom?

Yes, the Sierpinski space is an example of a space that satisfies the $T_0$ axiom but not the $T_1$ axiom. This space has two points, one of which is open and the other is not, so there is no way to separate the two points with disjoint open sets.

4. How are separation axioms related to the Hausdorff axiom?

The Hausdorff axiom, also known as the $T_2$ separation axiom, is the strictest of the five separation axioms. It requires that for any two distinct points in a space, there exist two disjoint open sets, each containing one of the points. This axiom implies all of the other separation axioms, so any space that satisfies the Hausdorff axiom automatically satisfies all of the other separation axioms as well.

5. What are some real-life applications of separation axioms?

Separation axioms are important in topology and can be applied to various real-world problems, such as network routing, data clustering, and image recognition. They also play a crucial role in the study of manifolds and other geometric structures.

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