Abstract
In this chapter we describe a technique that is used to refine a separable metrizable topology so as to produce a locally countable and locally compact new topology with various interesting pathologies. The main feature of the technique is that of arranging for some carefully selected sequences that have uncountably many common limit points in the old topology, to also have uncountably many common limit points relative to the new topology. This ensures that the new topology inherits some of the good properties of the old one.
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Notes
- 1.
To be pedantic, \(\mathcal {S}\) consists of all sequences (S, Q ∖ S, Q ∖ S, …) satisfying S ⊂ Q and \(| \mathop {\mathrm {cl}} \nolimits _\sigma (S) \cap \mathop {\mathrm {cl}} \nolimits _\sigma (Q \setminus S)|= \mathfrak {c}\).
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Charalambous, M.G. (2019). The van Douwen Technique for Constructing Counterexamples. In: Dimension Theory. Atlantis Studies in Mathematics, vol 7. Springer, Cham. https://doi.org/10.1007/978-3-030-22232-1_25
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