Math Is Fun Forum

blbl

Member

Registered: 2010-05-25

Posts: 4

separation axiom

verify the separation axiom "T₀ , T₁ " for (ℝ,cc) .
where cc is cocountable topology

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: separation axiom

For T_0, you want to find an open set that contains a but does not contain b, where a is not equal to b.  Are you having problems with this?

For T_1, remember that the sets you wish to separate two points with need not be open.  In fact, because the cocountable topology is not Hausdorff, it is actually required that they are not open.  With T_1, all that is required is that their closures not intersect.  With this is mind, your life is made much easier if you choose closed sets.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."