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a line segment (including its endpoints) is toplogically equivalent to a point.

consider S a nonempty totally ordered (ie a set with a relation <= such that <= is reflexive, transitive, x<=y and y<=x imply x=y, and every pair is comparable--i think that's what total order means anyway) set.

i'm suspecting that the subsets of S of the form [tex]G_{y}:=\left\{ x\in S:y\leq x\right\} [/tex] form a topology on S. oh, let's assume S has a [tex]\leq[/tex]-maximal element and minimal element to insure that S and Ø are in the topology. i think one may need zorn's lemma to prove that the union of open sets is open. for finite intersections of [tex]G_{y}[/tex]'s, take the max of the y's, call that max y' and then consider [tex]G_{y'}[/tex] which is i believe the intersection of the others.

now is S equipped with this topology homeomorphic to a point (with the concrete topology)? is it obviously not homeomorphic?

well, if there is a universal set, then let S=U. S turns into a totally ordered set with <= meaning injection. S has a minimum and maximum <= element, namely Ø and U, and so U would be topologically equivalent to 1={Ø}. perhaps another equation up there with euler's formula would be:

{Ø}=1~U

note to Self: 435.

consider S a nonempty totally ordered (ie a set with a relation <= such that <= is reflexive, transitive, x<=y and y<=x imply x=y, and every pair is comparable--i think that's what total order means anyway) set.

i'm suspecting that the subsets of S of the form [tex]G_{y}:=\left\{ x\in S:y\leq x\right\} [/tex] form a topology on S. oh, let's assume S has a [tex]\leq[/tex]-maximal element and minimal element to insure that S and Ø are in the topology. i think one may need zorn's lemma to prove that the union of open sets is open. for finite intersections of [tex]G_{y}[/tex]'s, take the max of the y's, call that max y' and then consider [tex]G_{y'}[/tex] which is i believe the intersection of the others.

now is S equipped with this topology homeomorphic to a point (with the concrete topology)? is it obviously not homeomorphic?

well, if there is a universal set, then let S=U. S turns into a totally ordered set with <= meaning injection. S has a minimum and maximum <= element, namely Ø and U, and so U would be topologically equivalent to 1={Ø}. perhaps another equation up there with euler's formula would be:

{Ø}=1~U

note to Self: 435.

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