In [9], R. S. Phillips gave a compactness criterion for subsets of a Banach space X, namely, that if (Ts) is a net of compact operators converging strongly to the identity then a bounded set A ⊆ X is relatively compact if and only if (Ts) converges uniformly on A to the identity. He then used this general result to give concrete compactness criteria in some specific spaces and to investigate compactness of operators. In this paper, we develop these ideas in two directions: firstly, to demonstrate that this method can be used as a unifying tool to derive many classical compactness criteria originally proved using other techniques, and secondly to extend the method from giving a criterion for compactness to giving a generalised measure of noncompactness. The conditions for compactness that result from this approach are not new (though they are possibly given in slightly more generality than can easily be found in the literature), but they all arise from a single, simple, and usually elementary, approach.

Throughout, vector operations applied to sets are all defined pointwise, so if A and B are subsets of a vector space, x is a vector and λ is a scalar, then

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The ball centred at x with radius r in a normed space X is denoted BX(x; r). More generally, if A is a subset of a normed space X then the r-neighbourhood BX(A; r) of A is defined by

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It is sometimes convenient to combine these notations, for example to use x+rBX(0; 1) in place of BX(x; r) or A+rBX(0; 1) in place of BX(A; r).

The ball measure of noncompactness in a normed space X will be denoted by βX: if A is a bounded subset of X then βX(A) is the infimum of r such that A can be covered with finitely many balls of radius r.

The bordism class of a self-transverse immersion f:Mn−k ℝn corresponds to an element α of the homotopy group πnΩ∞Σ∞MO(k). It is explained how the ℤ/2 Hurewicz image h(α)∈Hn(Ω∞Σ∞MO(k);ℤ/2) may be used to determine the characteristic numbers of the self-intersection manifolds Δr(f) of the immersion f.

Let T be a self-affine tile in ℝn defined by an integral expanding matrix A and a digit set D. The paper gives a necessary and sufficient condition for the connectedness of T. The condition can be checked algebraically via the characteristic polynomial of A. Through the use of this, it is shown that in ℝ2, for any integral expanding matrix A, there exists a digit set D such that the corresponding tile T is connected. This answers a question of Bandt and Gelbrich. Some partial results for the higher-dimensional cases are also given.

The singular homology groups of compact CW-complexes are finitely generated, but the groups of compact metric spaces in general are very easy to generate infinitely and our understanding of these groups is far from complete even for the following compact subset of the plane, called the Hawaiian earring:

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Griffiths [11] gave a presentation of the fundamental group of ℍ and the proof was completed by Morgan and Morrison [15]. The same group is presented as the free σ-product [smashp ]σℕℤ of integers ℤ in [4, Appendix]. Hence the first integral singular homology group H1(ℍ) is the abelianization of the group [smashp ]σℕℤ. These results have been generalized to non-metrizable counterparts ℍI of ℍ (see Section 3).

In Section 2 we prove that H1(X) is torsion-free and Hi(X) = 0 for each one-dimensional normal space X and for each i [ges ] 2. The result for i [ges ] 2 is a slight generalization of [2, Theorem 5]. In Section 3 we provide an explicit presentation of H1(ℍ) and also H1(ℍI) by using results of [4].

Throughout this paper, a continuum means a compact connected metric space and all maps are assumed to be continuous. All homology groups have the integers ℤ as the coefficients. The bouquet with n circles [xcup ]nj=1Cj is denoted by Bn. The base point (0, 0) of Bn is denoted by o for simplicity.

Consider the following infinite dimensional stochastic evolution equation over some Hilbert space H with norm [mid ]·[mid ]:

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It is proved that under certain mild assumptions, the strong solution Xt(x0)∈V[rarrhk ]H[rarrhk ]V*, t [ges ] 0, is mean square exponentially stable if and only if there exists a Lyapunov functional Λ(·, ·)[ratio ]H×R+→R1 which satisfies the following conditions:

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where [Lscr ] is the infinitesimal generator of the Markov process Xt and ci, ki, μi, i = 1, 2, 3, are positive constants. As a by-product, the characterization of exponential ultimate boundedness of the strong solution is established as the null decay rates (that is, μi = 0) are considered.