On the Reverse Mathematics of General Topology
Open Access
- Author:
- Mummert, Carl
- Graduate Program:
- Mathematics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- May 12, 2005
- Committee Members:
- Stephen George Simpson, Committee Chair/Co-Chair
John D Clemens, Committee Member
Alexander Nabutovsky, Committee Member
Martin Furer, Committee Member
Dmitri Yu Burago, Committee Member - Keywords:
- Reverse Mathematics
poset space
general topology
second-order arithmetic - Abstract:
- <p> This thesis presents a formalization of general topology in second-order arithmetic. Topological spaces are represented as spaces of filters on partially ordered sets. If <tt>P</tt> is a poset, let <tt>MF(P)</tt> be the set of maximal filters on <tt>P</tt>. Let <tt>UF(P)</tt> be the set of unbounded filters on <tt>P</tt>. If <tt>X</tt> is <tt>MF(P)</tt> or <tt>UF(P)</tt>, the topology on <tt>X</tt> has a basis <tt>{N<sub>p</sub> | p in P }</tt>, where <tt>N<sub>p</sub> = { F in X | p in F }</tt>. Spaces of the form <tt>MF(P)</tt> are called MF spaces; spaces of the form <tt>UF(P)</tt> are called UF spaces. A poset space is either an MF space or a UF space; a poset space formed from a countable poset is said to be countably based. The class of countably based poset spaces includes all complete separable metric spaces and many nonmetrizable spaces including the Gandy--Harrington space. All poset spaces have the strong Choquet property. </p> <p> This formalization is used to explore the Reverse Mathematics of general topology. The following results are obtained. </p> <p> <tt>RCA<sub>0</sub></tt> proves that countable products of countably based MF spaces are countably based MF spaces. The statement that every <tt>G<sub>delta</sub></tt> subspace of a countably based MF space is a countably based MF space is equivalent to <tt>Pi<sup>1</sup><sub>1</sub>CA<sub>0</sub></tt> over <tt>RCA<sub>0</sub></tt>. </p> <p> The statement that every regular countably based MF space is metrizable is provable in <tt>Pi<sup>1</sup><sub>2</sub>CA<sub>0</sub></tt> and implies <tt>ACA<sub>0</sub></tt> over <tt>RCA<sub>0</sub></tt>. The statement that every regular MF space is completely metrizable is equivalent to <tt>Pi<sup>1</sup><sub>2</sub>CA<sub>0</sub></tt> over <tt>Pi<sup>1</sup><sub>1</sub>CA<sub>0</sub></tt>. The corresponding statements for UF spaces are provable in <tt>Pi<sup>1</sup><sub>1</sub>CA<sub>0</sub></tt>, and each implies <tt>ACA<sub>0</sub></tt> over <tt>RCA<sub>0</sub></tt>. </p> <p> The statement that every countably based Hausdorff UF space is either countable or has a perfect subset is equivalent to <tt>ATR<sub>0</sub></tt> over <tt>ACA<sub>0</sub></tt>. <tt>Pi<sup>1</sup><sub>2</sub>CA<sub>0</sub></tt> proves that every countably based Hausdorff MF space has either countably many or continuum-many points; this statement implies <tt>ATR<sub>0</sub></tt> over <tt>ACA<sub>0</sub></tt>. The statement that every closed subset of a countably based Hausdorff MF space is either countable or has a perfect subset is equivalent over <tt>Pi<sup>1</sup><sub>1</sub>CA<sub>0</sub></tt> to the statement that <tt>aleph<sub>1</sub></tt> of <tt>L(A)</tt> is countable for every subset <tt>A</tt> of the natural numbers. </p>