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2 results

  • 9 - Topology

  • from Part III - Where Are the Paradoxes?
  • Zach Weber, University of Otago, New Zealand
  • Book:
    Paradoxes and Inconsistent Mathematics
    Published online:
    08 October 2021
    Print publication:
    21 October 2021, pp 256-282

  • Summary

    This chapter develops some topology, the abstract geometry ofcloseness, that manages to capture properties of nearness withoutany appeal to distance. The previous chapter studied the shape ofthe linear continuum, focusing on what happens when one tries to cutor tear it in two. This chapter offers a qualitative generalizationof these ideas, about continuity and connectedness, but now withoutany metrics. The focus is on closed sets, boundaries, andeventually, continuous transformations and their fixed points,bringing ideas from analysis back to set theory. Concluding theoremsare proven about retractions and some lemmas about absolutelydisconnected space, leading to a parameterized version ofBrouwer’s fixed point theorem.

  • SHEAF RECURSION AND A SEPARATION THEOREM

  • NATHANAEL LEEDOM ACKERMAN
  • Journal:
    The Journal of Symbolic Logic / Volume 79 / Issue 3 / September 2014
    Published online by Cambridge University Press:
    18 August 2014, pp. 882-907
    Print publication:
    September 2014

  • Define a second order tree to be a map between trees (with fixed codomain). We show that many properties of ordinary trees have analogs for second order trees. In particular, we show that there is a notion of “definition by recursion on a well-founded second order tree” which generalizes “definition by transfinite recursion”. We then use this new notion of definition by recursion to prove an analog of Lusin’s Separation theorem for closure spaces of global sections of a second order tree.