The convergence-theoretic approach to weakly first countable spaces and symmetrizable spaces

A.1 Set-theoretic conventions

If X is a set, we denote by 2^X its powerset, by [X]^<∞ the set of finite subsets of X and by [X]^ω the set of countable subsets of X. If 𝓐 ⊂ 2^X, we write

A family 𝓕 of non-empty subsets of X is called a filter if 𝓕 = 𝓕^∩ = 𝓕^↑. We denote by 𝔽X the set of filters on X. Note that 2^X is the only family 𝓐 satisfying 𝓐 = 𝓐^↑ = 𝓐^∩ that has an empty element. Thus we sometimes call {∅}^↑ = 2^X the degenerate filter onX. The set 𝔽X is ordered by 𝓕 ≤ 𝓖 if for every F ∈ 𝓕 there is G ∈ 𝓖 with G ⊂ F. Maximal elements of 𝔽X are called ultrafilters and 𝓕 ∈ 𝔽X is an ultrafilter if and only if 𝓕 = 𝓕^#. We denote by 𝕌X the set of ultrafilters on X.

A sequence {xn}n=1∞ on a set X induces a filter

A filter that is induced by some sequence is called sequential filter. We denote by 𝔼X the set of sequential filters on X. A filter 𝓕 ∈ 𝔽X is called Fréchet if

Of course, every countably based filter, that is, a filter with a countable filter-base, is in particular a Fréchet filter.

A.2 Convergence

A convergenceξ on a set X is a relation between 𝔽X and X, denoted

whenever (x, 𝓕) ∈ ξ (and we then say that 𝓕 converges toxforξ), satisfying the following two conditions:

for every filters 𝓕 and 𝓖 on X, and every x ∈ X.

The pair (X, ξ) is then called a convergence space. We denote by limξ−(x) the set of filters that converge to x for ξ.

Every topology can be seen as a convergence. Indeed, if τ is a topology and 𝓝_τ(x) denotes the neighborhood filter of x for τ, then

defines a convergence that completely characterizes τ. Hence, we do not distinguish between a topology τ and the convergence it induces. A convergence is called Hausdorff if the cardinality of lim 𝓕 is at most one, for every filter 𝓕. Of course, a topology is Hausdorff in the usual topological sense if and only if it is in the convergence sense. A point x of a convergence space (X, ξ) is isolated if limξ−(x) = {{x}^↑}. A prime convergence is a convergence with at most one non-isolated point.

Given two convergences ξ and θ on the same set X, we say that ξ is finer thanθ or that θ is coarser thanξ, in symbols ξ ≥ θ, if lim_ξ𝓕 ⊂ lim_θ𝓕 for every 𝓕 ∈ 𝔽X. With this order, the set 𝓒(X) of convergences on X is a complete lattice whose greatest element is the discrete topology and least element is the antidiscrete topology, and for which, given Ξ ⊂ 𝓒(X),

A base for a convergence space (X, ξ) is a family 𝓑 of subsets of X such that for every x ∈ X and every 𝓕 with x ∈ lim_ξ𝓕, there is a filter 𝓖 with a filter-base composed of elements of 𝓑 with x ∈ lim_ξ𝓖 and 𝓕 ≤ 𝓖. If this property is satisfied only for a specific x, then 𝓑 is a local base atx.

A.3 Continuity, initial and final constructions

A map f between two convergence spaces (X, ξ) and (Y, σ) is continuous if for every 𝓕 ∈ 𝔽X and x ∈ X,

where

Consistently with [7], we denote by |ξ| the underlying set of a convergence ξ, and, if (X, ξ) and (Y, σ) are two convergence spaces, we often write f : |ξ| → |σ| instead of f : X → Y even though one may see it as improper since many different convergences have the same underlying set. This allows to talk about the continuity of f : |ξ| → |σ| without having to repeat for what structures.

Given a map f : |ξ| → Y, there is the finest convergence fξ on Y making f continuous (from ξ), and given f : X → |σ|, there is the coarsest convergence f⁻σ on X making f continuous (to σ). The convergences fξ and f⁻σ are called final convergence forfandξ and initial convergence forf andσ respectively. Note that

If A ⊂ |ξ|, the induced convergence byξonA, or subspace convergence, is i⁻ξ, where i : A → |ξ| is the inclusion map. If ξ and τ are two convergences, the product convergenceξ × τ on |ξ| × |τ| is the coarsest convergence on |ξ| × |τ| making both projections continuous, that is,

where p_ξ : |ξ| × |τ| → |ξ| and p_τ : |ξ| × |τ| → |τ| are the projections defined by p_ξ(x, y) = x and p_τ(x, y) = y respectively.

A.4 Topologies and pretopologies

In fact, the category Top of topological spaces and continuous maps is a reflective subcategory of Conv and the corresponding reflector T, called topologizer, associates with each convergence ξ on X its topological modification Tξ, which is the finest topology on X among those coarser than ξ (in Conv). Concretely, Tξ is the topology whose closed sets are the subsets of |ξ| that are ξ-closed, that is, subsets C satisfying

A subset O of |ξ| is ξ-open if its complement is closed, equivalently if

The neighborhood filter 𝓝_ξ(x) of x for ξ is the neighborhood filter of x for the topology Tξ, that is, the filter generated by the family of ξ-open sets containing x. Because Top is a (full) reflective subcategory of Conv, Top is closed under initial constructions so that a subspace of a topological convergence space is topological and a product of topological convergence spaces is topological. However, the convergence induced by Tξ on a subset A and T(i⁻ξ) do not need to coincide, and similarly, the topological modification of a product generally does not coincide with the product of the topological modifications.

Given a convergence ξ and x ∈ |ξ|, the filter

is called the vicinity filter ofx for ξ. In general, 𝓥_ξ(x) does not need to converge to x for ξ. If x ∈ lim_ξ𝓥_ξ(x) for all x ∈ |ξ|, we say that ξ is a pretopology or is pretopological. Note that a local base at x of a pretopology is necessarily a filter-base of 𝓥_ξ(x).

The category PrTop of pretopological spaces and continuous maps is a (full) reflective subcategory of Conv. The corresponding reflector S₀, called pretopologizer, associates with each convergence ξ its pretopological modification S₀ξ, which is the finest among the pretopologies coarser than ξ. Explicitly, x ∈ lim_S0ξ𝓕 if 𝓕 ≥ 𝓥_ξ(x) so that 𝓥_ξ(x) = 𝓥_S0ξ(x). Topologies are in particular pretopological, so that S₀ ≥ T. Moreover,

and ξ is topological if and only if 𝓥_ξ(x) has a filter-base composed of open sets, in which case 𝓥_ξ(x) = 𝓝_ξ(x). We distinguish between the inherence

of A ⊂ |ξ|} and its interior

In contrast to T, the reflector S₀ does commute with initial convergence (so that the pretopological modification of an induced convergence is the convergence induced by the pretopological modification), but not with suprema, hence not with products.

A.5 Sequential, Fréchet, and strongly Fréchet convergences

With a convergence ξ, we associate its sequentially based modification Seqξ defined by

A useful immediate consequence of [4: Corollary 10] (that can be traced all the way back to [14], but [4] gives a formulation more consistent with our terminology) is:

In the same vein, letting 𝔽₁X denote the set of filters with a countable filter-base, we can associate to each convergence ξ its modification of countable character I₁ξ defined by

Both Seq and I₁ are concrete coreflectors from Conv to the full categories of sequentially based and convergences of countable character respectively.

Recall that a topological space is sequential if every sequentially closed subset (that is, subset that contains the limit point of every sequence on it) is closed. It is easily seen that a subset of a topology ξ is sequentially closed if and only if it is Seq ξ-closed, so that, ξ is sequential if and only if ξ = T Seq ξ. As σ ≤ Seq σ for every convergence σ, the inequality ξ ≤ T Seq ξ is true for every topology ξ. Hence if ξ = Tξ then

and it turns out that

and we take one or the other of these inequalities as a definition of a sequential convergence.

A topological space X is Fréchet if for every x ∈ X and A ⊂ X, if x ∈ clA then there is a sequence on A converging to x. It is easily seen (e.g., [15][2]) that if ξ is a topology then

and we take either one of these inequalities as a definition of a Fréchet convergence. Note that a pretopology is Fréchet if and only if each vicinity filter is a Fréchet filter in the sense of (A.1).

The paratopologizer S₁ is defined by

and defines a concrete reflector. A fixed point of S₁ is called a paratopology.

A topological space X is strongly Fréchet if whenever x ∈ adh𝓗 for a countably based filter 𝓗 on X, there is a sequential filter (x_n)_n with (x_n)_n ≥ 𝓗 and x ∈ lim(x_n)_n. It is easily seen (e.g., [2]) that if ξ is a topology then

and we take either one of these inequalities as a definition of a strongly Fréchet convergence.

A.6 Properties of concrete functors

Several other topological properties can be characterized with the aid of a functorial inequality; see [2, 7]. All functors considered here (in particular T, S₀, S₁, Seq and I₁ ) are concrete endofunctors of Conv, and as such each satisfy the following properties of a modifier F acting on convergence spaces (for all ξ,θ and f):

All five functors T, S₀, S₁, Seq and I₁ are also idempotent, that is, satisfy F(Fξ) = Fξ for all ξ. The reflectors T, S₀ and S₁ are additionally contractive (Fξ ≤ ξ for all ξ) and the coreflectors Seq and I₁ are additionally expansive (Fξ ≥ ξ for all ξ).

A.7 Compactness and local compactness

A subset K of a convergence space is ξ-compact if lim_ξ𝓤 ∩ K ≠ ∅ for every ultrafilter 𝓤 on K, and countably compact if every countably based filter 𝓗 with K ∈ 𝓗^#, there is an ultrafilter 𝓤 ≥ 𝓗 with lim_ξ𝓤 ∩ K ≠ ∅. Given two convergences ξ and σ on the same set, we say that ξ is locally (countably)σ-compact if every ξ-convergent filter has a (countably) σ-compact element.