Topology question

[Cratylus]

Given the discrete ,indiscrete ,cotopology or Seirenpiski topology that is defined on {0,1}
Which is true in each topology of the below
1. T_0 space
2. Hausdorff
3 T_1 space

ex is Hausdorff space true in the discrete topology,..
l know the defs but don’t know how to go about this one

 

[HallsofIvy]

Okay, first tell us what you beieve those definitions are. Then we can tell you whether they are correct and, if so, guide you in applying them.

 

[Cratylus]

Okay, first tell us what you beieve those definitions are. Then we can tell you whether they are correct and, if so, guide you in applying them.

By the defs ,l was referring to the spaces.
As to the topolo gives, l apply each to the set {0,1} ,right?
A T0 space is a topological space in which every pair of distinct points is topologically distinguishable

A T1 space is a topological space in which, for every pair of distinct points, each has a neighborhood not containing the other point

A T2 space is a topological spacewhere for any two distinct points there exist neighbourhoods of each which are disjoint from each other.

l think Hausdorff is true in cofinite and discrete topology
l think T0 is true on discrete and cofinite
Cant say anything about T1

 

[pka]

Given the discrete ,indiscrete ,cotopology or Seirenpiski topology that is defined on {0,1}
Which is true in each topology of the below
1. T_0 space
2. Hausdorff
3 T_1 space

It seems to me as if you have the idea of the axioms of point-separation backwards.
For example a Hausdroff space is a \(\bf T_2\) space. That is, if \(x~\&~y\) are two points then there exists
open sets \(U~\&~V\) such that \(x\in U~.~y\in V~\&~U\cap V=\emptyset\).

 

[Cratylus]

It seems to me as if you have the idea of the axioms of point-separation backwards.
For example a Hausdroff space is a \(\bf T_2\) space. That is, if \(x~\&~y\) are two points then there exists open sets U and V such that x$\in$ U and y$\in$ V and U $\cap$V=0
open sets \(U~\&~V\) such that \(x\in U~.~y\in V~\&~U\cap V=\emptyset\).

l got them off the Wiki. How do l got them backwards?
From A first course in Topology,” lf x & y are two distinct points in a space then
there exists open sets U and V such that x [MATH]\in U[/MATH] and y [MATH]\in [/MATH] V and U[MATH]\cap[/MATH] V={}
( A space of this definition is called a [MATH]T_2[/MATH] space or a Hausdorff space)
pg 62

So the topologies that l have to check for each separation axiom are
1.discrete
2.indiscrete
3.cofinite
4. indiscrete topology on {0,1}
{0,{01}}
5. Discrete topology on {0,1}
{0,{0},{1},{0,1}}
6 .S={0,{0},{0,1}}
seirenpinski {S,{0,1}}

and state for each axiom if is true or false
so for each topology there are 3 checks to make