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We demonstrate that many properties of topological spaces connected with the notion of resolvability are preserved by the relation of similarity between topologies. Moreover, many of them can be characterised by the properties of the algebra of sets with nowhere dense boundary and the ideal of nowhere dense sets. We use these results to investigate whether a given pair of an algebra and an ideal is topological.

It is an open question whether every derivation of a Fréchet GB$^{\ast }$-algebra $A[{\it\tau}]$ is continuous. We give an affirmative answer for the case where $A[{\it\tau}]$ is a smooth Fréchet nuclear GB$^{\ast }$-algebra. Motivated by this result, we give examples of smooth Fréchet nuclear GB$^{\ast }$-algebras which are not pro-C$^{\ast }$-algebras.

Let$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}K$be a locally compact hypergroup endowed with a left Haar measure and let$L^1(K)$be the usual Lebesgue space of$K$with respect to the left Haar measure. We investigate some properties of$L^1(K)$under a locally convex topology$\beta ^1$. Among other things, the semireflexivity of$(L^1(K), \beta ^1)$and of sequentially$\beta ^1$-continuous functionals is studied. We also show that$(L^1(K), \beta ^1)$with the convolution multiplication is always a complete semitopological algebra, whereas it is a topological algebra if and only if$K$is compact.

The variety of topological groups generated by the class of all abelian kω-groups has been shown to equal the variety of topological groups generated by the free abelian topological group on [0, 1]. In this paper it is proved that the free abelian topological group on a compact Hausdorff space X generates the same variety if and only if X is not scattered.

Let C*(E) = C*(se, pv) be the graph C*-algebra of a directed graph E = (E0, E1) with the vertices E0 and the edges E1. We prove that if E is a finite graph (possibly with sinks) and φE: C*(E) → C*(E) is the canonical completely positive map defined by then Voiculescu's topological entropy ht(φE) of φE is log r(AE), where r(AE) is the spectral radius of the edge matrix AE of E. This extends the same result known for finite graphs with no sinks. We also consider the map φE when E is a locally finite irreducible infinite graph and prove that , where the supremum is taken over the set of all finite subgraphs of E.

The problem of metrising abstract topological spaces constitutes one of the major themes of topology. Since, for each new significant class of topological spaces this question arises, the problem is always current. One of the famous metrisation problems is the Normal Moore Space Conjecture. It is known from relatively recent work that one must add special conditions in order to be able to get affirmative results for this problem. In this paper we establish such special conditions. Since these conditions are characterised by local simplicity and global coherence they are referred to in this paper generically as “abstract topological manifolds.” In particular we establish a generalisation of a classical development of Bing, giving a proof which is complete in itself, not depending on the result or arguments of Bing. In addition we show that the spaces recently developed by Collins designated as “W satisfying open G(N)” are metrisable if they are locally separable and locally connected and regular. Finally, we establish a new necessary and sufficient condition for spaces to be metrisable.

We study the class of Tychonoff topological spaces such that the free Abelian topological group A(X) is reflexive (satisfies the Pontryagin-van Kampen duality). Every such X must be totally path-disconnected and (if it is pseudocompact) must have a trivial first cohomotopy group π1(X). If X is a strongly zero-dimensional space which is either metrisable or compact, then A(X) is reflexive.

This thesis examines topological states of matter from the perspective of noncommutative geometry and KK-theory. Examples of such topological states of matter include the quantum Hall effect and topological insulators. For the quantum Hall effect, we consider a continuous model and show that the Hall conductance can be expressed in terms of the index pairing of the Fermi projection of a disordered Hamiltonian with a spectral triple encoding the geometry of the sample’s momentum space. The presence of a magnetic field means that noncommutative algebras and methods must be employed. Higher dimensional analogues of the quantum Hall system are also considered, where the index pairing produces the ‘higher-dimensional Chern numbers’ in the continuous setting. Next we consider a discrete quantum Hall system with an edge. We show that topological properties of observables concentrated at the boundary can be linked to invariants from a boundary-free model via the Kasparov product. Hence we obtain the bulk-edge correspondence of the quantum Hall effect in the language of KK-theory. Finally we consider topological insulators, which come from imposing (possibly anti-linear) symmetries on condensed-matter systems and studying the invariants that are protected by these symmetries. We show how symmetry data can be linked to classes in real or complex KK-theory. Finally we prove the bulk-edge correspondence for topological insulator systems by linking bulk and edge systems using the Kasparov product in KKO-theory.

We survey the present trends in theory of universal arrows to forgetful functors from various categories of topological algebra and functional analysis to categories of topology and topological algebra. Among them are free topological groups, free locally convex spaces, free Banach-Lie algebras, and more. An accent is put on the relationship of those constructions with other areas of mathematics and their possible applications. A number of open problems is discussed; some of them belong to universal arrow theory, and other may become amenable to the methods of this theory.

We characterise in terms of free topological groups those Banach-Lie algebras with finite-dimensional centre coming from Lie groups.

We give a characterization of monoidal closed structures, “determined by systems of filters” on the category of topological spaces and continuous maps. The method we use to introduce suitable topologies on the product set X × Y of spaces X and Y, and on the set of all continuous maps from X to Y, is essentially that of Wilker.

Every multiplicative linear functional on a pseudocomplete locally convex algebra satisfying the “sequential” property of Husain and Ng is bounded (a topological algebra is called “sequential” if every null sequence contains an element whose powers converge to zero). Characterizations of such algebras are given, with some examples.

I this paper we extend such dynamical concepts as weak almost periodicity and simple equicontinuity of topological dynamics to the context of groups of *-automorphisms on a C*-algebra with unit. If (A, G) is a C* flow and if S(A) is the state space of A, we show that (A, G) is weakly almost periodic if and only if (S(A), G) is weakly almost periodic, and that, if A is a von Neumann algebra, then A is G-finite if and only if G is simple equicontinuous on the unit ball of A with respect to the weak *-topology.

M.I.Graev has shown that subgroups of free topological groups need not be free. Brown and Hardy, however, have proved that any open subgroup of the free topological group on a kw-space is again a free topological group: indeed, this is true for any closed subgroup for which a Schreier transversal can be chosen continuously. This note provides a proof of this result more direct than that of Brown and Hardy. An example is also given to show that the condition stated in the theorem is not a necessary condition for freeness of a subgroup. Finally, a sharpened version of a particular case of the theorem is obtained, and is applied to the preceding example.

This paper characterizes topological spaces which have an Abian structure - a uniform-like structure of the covering type.

In this paper neighbourhood lattices are developed as a generalisation of topological spaces in order to examine to what extent the concepts of “openness”, “closedness”, and “continuity” defined in topological spaces depend on the lattice structure of P(X), the power set of X.A general pre-neighbourhood system, which satisfies the poset analogues of the neighbourhood system of points in a topological space, is defined on an ∧-semi-lattice, and is used to define open elements. Neighbourhood systems, which satisfy the poset analogues of the neighbourhood system of sets in a topological space, are introduced and it is shown that it is the conditionally complete atomistic structure of P(X) which determines the extension of pre-neighbourhoods of points to the neighbourhoods of sets.The duals of pre-neighbourhood systems are used to generate closed elements in an arbitrary lattice, independently of closure operators or complementation. These dual systems then form the backdrop for a brief discussion of the relationship between preneighbourhood systems, topological closure operators, algebraic closure operators, and Čech closure operators.Continuity is defined for functions between neighbourhood lattices, and it is proved that a function f: X → Y between topological spaces is continuous if and only if corresponding direct image function between the neighbourhood lattices P(X) and P(Y) is continuous in the neighbourhood sense. Further, it is shown that the algebraic character of continuity, that is, the non-convergence aspects, depends only on the properites of pre-neighbourhood systems. This observation leads to a discussion of the continuity properties of residuated mappings. Finally, the topological properties of normality and regularity are characterised in terms of the continuity properties of the closure operator on a topological space.

In this paper we prove a coincidence theorem in not necessarily locally convex topological vector spaces, which contains, as a special case, a coincidence theorem proved by Felix Browder. As an application, a result about the existence of maximal elements is obtained.

Each continuous positive linear form on a commutative locally m-convex *-algebra E with a bounded approximate identity, accepts an integral representation on the hermitian spectrum (hermitian characters) of E. An alternative form of the latter is also obtained. The presented results constitute an abstract form of the Bochner-Weil-Raikov theorem within the frame of topological *-algebras.

Aspects of duality relating to compact totally disconnected universal algebras are considered. It is shown that if P is a ““basic“ set of injectives in a variety of compact totally disconnected algebras then the category P of P-copresentable objects is in duality with the class of all G-copresentable algebras on P, where G: P → Ens is the forgetful functor and an algebra is taken to mean a finite-product-preserving functor from P to Ens.

Let B be a topological algebra with Fréchet space topology, A an algebra with locally convex topology and an algebra of formal power series over A in n commuting indeterminates which carries a Fréchet space topology. In a previous paper the author showed, for the case n = 1, that a homomorphism of B into whose range contains polynomials is necessarily continuous provided the coordinate projections of into A satisfy a certain equicontinuity condition. This result is here extended to the case of general n, and also to weaker topological assumptions.

Let A be an algebra of formal power series in one indeterminate over the complex field, D a derivation on A. It is shown that if A has a Fréchet space topology under which it is a topological algebra, then D is necessarily continuous provided the coordinate projections satisfy a certain equicontinuity condition. This condition is always satisfied if A is a Banach algebra and the projections are continuous. A second result is given, with weaker hypothesis on the projections and correspondingly weaker conclusion.

Let X be a commutative locally convex Hausdorff topological algebra with identity over a non-trivially valued field F. Let Mc denote the continuous nontrivial homomorphisms of X into F and M the set of all maximal ideals of X. If the spectrum of each element x in X is the set of scalars {f(x) | f ∈ Mc}, it is shown that the singular elements of X are weakly dense in X if and only if M is an infinite set.

In this paper we prove two results relating the cardinalities of discrete subsets to those of dense subsets of a topological space. In particular, we show that(1) closed discrete subsets of separable normal spaces are countable, and(2) discrete subsets of separable regular spaces have cardinalities at most that of the continuum.

It has been observed by Banaschewski [1], Proposition 1, that, in the notation of the author's paper [2], if P ⊆ FU is hereditary and finitely productive. This fact does not require the use of injectives as in [2].

Theorem 1 and Corollary 1 of the original paper [Bull. Austral. Math. Soc. 25 (1982), 99–101] are false unless, for example, we assume the Generalised Continuum Hypothesis. The proof of Theorem 1 correctly shows that exp(|c|) ≤ exp(|D|) but one cannot then deduce that the cardinality of C is at most that of D. All one can deduce is that the cardinality of C is less than exp(|D|).