On the generalized Sierpinski spaceIn Sierpiński topology the open sets are linearly ordered by set inclusion, i.e. If S = { 0 , 1 }, then the Sierpiński topology on S is the collection { ϕ , { 1 } , { 0 , 1 } } such that ϕ ⊂ { 0 } ⊂ { 0 , 1 } we can generalize it by defining a topology analogous to Sierpiński topology with nested open sets on any arbitrary non-empty set as follows: Let X be a non-empty set and I a collection of some nested subsets of X indexed by a linearly ordered set ( Λ , ≤ ) such that I always contains the void set ϕ and the whole set X, i.e. I = { ∅ , A λ , X : A λ ⊂ X , λ ∈ Λ }such that A μ ⊂ A ν whenever μ ≤ ν.Then it is easy to show that I qualifies as a topology on X.My questions are:(1) Is there a name for such a topology in general topology literature?(2) Is there any research paper studying such type of compact, non-Hausdorff and connected chain topologies?

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On the generalized Sierpinski spaceIn Sierpiński topology the open sets are linearly ordered by set inclusion, i.e. If   S  =  {  0  ,  1  }, then the Sierpiński topology on S is the collection   {      ϕ    ,  {  1  }  ,  {  0  ,  1  }  } such that   ϕ  ⊂  {  0  }  ⊂  {  0  ,  1  } we can generalize it by defining a topology analogous to Sierpiński topology with nested open sets on any arbitrary non-empty set as follows: Let X be a non-empty set and I a collection of some nested subsets of X indexed by a linearly ordered set   (  Λ  ,  ≤  ) such that I always contains the void set       ϕ   and the whole set X, i.e.  I  =  {  ∅  ,      A    λ    ,  X  :      A    λ    ⊂  X  ,  λ  ∈  Λ  }such that       A    μ    ⊂      A    ν   whenever   μ  ≤  ν.Then it is easy to show that I qualifies as a topology on X.My questions are:(1) Is there a name for such a topology in general topology literature?(2) Is there any research paper studying such type of compact, non-Hausdorff and connected chain topologies?

Lokasckf · Accepted Answer

Step 1This was already pointed out in comments, but I&#039;m collecting this into an answer. (In fact, this is closer to a longer comment than to an answer, but the details can be elaborated better here than in a short comment.)Both examples here deal with your construction in the special case that   X  =  Λ. And in both cases we work with the sets of the form   (  −  ∞  ,  a  )  =  {  x  ∈  Λ  ;  x  &amp;lt;  a  },  which is similar to lower topology.Step 2Suppose that we have   X  =  Λ and       A    λ    =  (  −  ∞  ,  λ  )  =  {  x  ∈  Λ  ;  x  &amp;lt;  λ  }.This means that for each bounded subset   D  ⊆  Λ there exists a   λ  such that       ⋃          μ      ∈      D            A    μ    =      A    λ  . It is relatively easy to show that the last condition is equivalent to   λ  =      sup          μ      ∈      D        μ. I.e. the linear order would have to be complete.A counterexample suggested by Niels is   Λ  =  X  =      Q  . For example if   (      q    n    ) is an increasing sequence of rational numbers such that       q    n    →      2  , then   ⋃  (  −  ∞  ,      q    n    )  =  (  −  ∞  ,      2    ), which is not of the form       A    λ  .There are several possibilities how to circumvent this. E.g. you could take all downward closed sets as open. Or you could take   {  ∅  ,  X  ,      A    λ    ;  λ  ∈  Λ  } as a base for the topology you want to generate.Step 3Even if this is a topology, it need not necessarily be compact.Let us take   X  =  Λ  =      Z  .Again       A    λ    =  {  x  ∈      Z    ;  x  &amp;lt;  λ  }  =  (  −  ∞  ,  λ  )  ∩      Z   I.e.       A    λ  &#039;s are down-sets of the linearly ordered set       Z  .Now if C is open cover of X then, for each   n  ∈      Z  , the cover C must contain some       A    λ   with   λ  &amp;gt;  n. his shows that every open cover is infinite.And if we take       C    =  {      A    n    ;  n  ∈      Z    }  =  {  (  −  ∞  ,  n  )  ∩      Z    ;  n  ∈      Z    }, we get an open cover which does not have a finite subcover.

(1) Is there a name for such a topology in general topology literature? (2) Is there any research paper studying such type of compact, non-Hausdorff and connected chain topologies?

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