Ultrafilter orders on chainable continua

Speaker(s)

Julia Ścisłowska

Affiliation

Doctoral School of Exact and Natural Sciences UW

Language of the talk

English

Date

March 5, 2025, 4:15 p.m.

Room

room 5050

Seminar

Topology and Set Theory Seminar

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The talk will be based on the results from my master’s thesis ,,Linear orders on chainable  continua", prepared under supervision of prof. Witold Marciszewski. The thesis is devoted to study families of ultrafilter orders on a given chainable continuum X (such as e.g. arc, the Warsaw sine curve, the Knaster continuum etc.). These orders depend on a fixed sequence of chains, covering X (obtained from chainability of X), and on fixed nonprincipal ultrafilter on N. Alternatively ultrafilter orders may be defined using representation of X as an inverse limit of a sequence of arcs and a fixed nonprincipal ultrafilter on N.

 During my talk I will introduce the notion of an ultrafilter order on a chainable continuum and present some results concerning ultrafilter orders on various chainable continua. In particular, I will discuss the theorems stating that exist exactly 2 distinct ultrafilter orders on any arc (i.e. space homeomorphic to [0, 1]), exactly 4 distinct ultrafilter orders on the Warsaw sine-curve (which is the closure of the set {(x, sin( 1/x )) : x ∈ (0, 2/3π ]}) and exactly continuum distinct ultrafilter orders on a certain chainable continuum, which has infinitely many arc components. I will also show that there exist 2^{continuum} distinct ultrafilter orders on the Knaster continuum and discuss some topological properties of the Knaster continuum equipped with an order topology generated by a certain ultrafilter order.

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