Every topological space is metrisable once the symmetry axiom is abandoned and the codomain of the metric is allowed to take values in a suitable structure tailored to fit the topology (and every completely regular space is similarly metrisable while retaining symmetry). This result was popularised in 1988 by Kopperman, who used value semigroups as the codomain for the metric, and restated in 1997 by Flagg, using value quantales. In categorical terms, each of these constructions extends to an equivalence of categories between the category Top and a category of all L-valued metric spaces (where L ranges over either value semigroups or value quantales) and the classical \epsilon-\delta notion of continuous mappings. Thus, there are (at least) two metric formalisms for topology, raising the questions: 1) is any of the two actually useful for doing topology? and 2) are the two formalisms equally powerful for the purposes of topology? After reviewing Flagg's machinery I will attempt to answer the former affirmatively and the latter negatively. In more detail, the two approaches are equipotent when it comes to point-to-point topological consideration, but only Flagg's formalism captures 'higher order' topological aspects correctly, however at a price; there is no notion of product of value quantales. En route to establishing Flagg's formalism as convenient, it will be shown that both fine and coarse variants of homology and homotopy arise as left and right Kan extensions of genuinely metrically constructed functors, and a topologically relevant notion of tensor product of value quantales, a surrogate for the non-existent products, will be described.

# Past Analytic Topology in Mathematics and Computer Science

Nash-Williams showed that the collection of locally finite trees under the topological minor relation results in a BQO. Naturally, two interesting questions arise:

1. What is the number \lambda of topological types of locally finite trees?

2. What are the possible sizes of an equivalence class of locally finite trees?

For (1), clearly, \omega_0 \leq \lambda \leq c and Matthiesen refined it to \omega_1 \leq \lambda \leq c. Thus, this question becomes non-trivial in the absence of the Continuum Hypothesis. In this paper we address both questions by showing - entirely within ZFC - that for a large collection of locally finite trees that includes those with countably many rays:

- \lambda = \omega_1, and

- the size of an equivalence class can only be either 1 or c.

A main part of the proof uses forcing to establish a Ramsey theorem on a new type of tree, though the result holds in ZFC. The space of such trees almost forms a topological Ramsey space.

The talk will focus on five items:

Theorem 1. It is ZFC-independent whether every locally compact, $\omega_1$-compact space of cardinality $\aleph_1$ is the union of countably many countably compact spaces.

Problem 1. Is it consistent that every locally compact, $\omega_1$-compact space of cardinality $\aleph_2$ is the union of countably many countably compact spaces?

[`$\omega_1$-compact' means that every closed discrete subspace is countable. This is obviously implied by being the union of countably many countably compact spaces, but the converse is not true.]

Problem 2. Is ZFC enough to imply that there is a normal, locally countable, countably compact space of cardinality greater than $\aleph_1$?

Problem 3. Is it consistent that there exists a normal, locally countable, countably compact space of cardinality greater than $\aleph_2$?

The spaces involved in Problem 2 and Problem 3 are automatically locally compact, because by "space" I mean "Hausdorff space" and so regularity is already enough to give every point a countable countably compact (hence compact) neighborhood.

Theorem 2. The axiom $\square_{\aleph_1}$ implies that there is a normal, locally countable, countably compact space of cardinality $\aleph_2$.

This may be the first application of $\square_{\aleph_1}$ to construct a topological space whose existence in ZFC is unknown.