Abstract:
In this paper the notation of base-mesocompactness is introduced and the following results are mainly obtained:(1) Let X be base-mesocompact and X' an F σ subset of X.If X is normal,then X' is base-mesocompact relative to X.(2) Let f:X → Y be a base-mesocompact mapping,ω(X) be a regular cardinality of X and ω(X) ≥ ω(Y).If Y is base-mesocompact and regular,then X is base-mesocompact.(3) Let f:X → Y be a closed lindelof mapping with regular domain and regular range.If Y is base-mesocompact,then X is base-mesocompact.(4) Let X be base-mesocompact.If Y is locally compact and base-mesocompact,then X × Y is base-mesocompact.Keywords:
Base (topology)
Base change
Cardinality (data modeling)
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$X$ is a (topological) space, the $n$th finite subset space of $X$, denoted by $X(n)$, consists of $n$-point subsets of $X$ (i.e., nonempty subsets of cardinality at most $n$) with the quotient topology induced by the unordering map $q:X^n\to X(n)$, $(x_1,\cdots,x_n)\mapsto\{x_1,\cdots,x_n\}$. That is, a set $A\subset X(n)$ is open if and only if its preimage $q^{-1}(A)$ is open in the product space $X^n$. Given a space $X$, let $H(X)$ denote all homeomorphisms of $X$. For any class of homeomorphisms $C\subset H(X)$, the $C$-geometry of $X$ refers to the description of $X$ up to homeomorphisms in $C$. Therefore, the topology of $X$ is the $H(X)$-geometry of $X$. By a ($C$-) geometric property of $X$ we will mean a property of $X$ that is preserved by homeomorphisms of $X$ (in $C$). Metric geometry of a space $X$ refers to the study of geometry of $X$ in terms of notions of metrics (e.g., distance, or length of a path, between points) on $X$. In such a study, we call a space $X$ metrizable if $X$ is homeomorphic to a metric space. Naturally, $X(n)$ always inherits some aspect of every geometric property of $X$ or $X^n$. Thus, the geometry of $X(n)$ is in general richer than that of $X$ or $X^n$. For example, it is known that if $X$ is an orientable manifold, then (unlike $X^n$) $X(n)$ for $n>1$ can be an orientable manifold, a non-orientable manifold, or a non-manifold. In studying geometry of $X(n)$, a central research question is If $X$ has geometric property $P$, does it follow that $X(n)$ also has property $P$?. A related question is If $X$ and $Y$ have a geometric relation $R$, does it follow that $X(n)$ and $Y(n)$ also have the relation $R$?. (Truncated)
Metrization theorem
Cardinality (data modeling)
Manifold (fluid mechanics)
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Hyperspace
Limit point
Compactification (mathematics)
Discrete space
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A real-valued function f on a topological space X is defined to be upper (lower) semi-continuous if the set { x: f(x) _ XI} (resp. {x: f(x) ?X}) is closed in X for each real number X [3, p. 101]. This notion has been generalized to a function from a topological space into some set of subsets of another topological space (cf. Hahn [2, p. 148] or Michael [4, p. 179]). More precisely, letting a be some collection of nonempty subsets of Y, we say that a function 4> from X to a is an upper (lower) semi-continuous carrier from X to a if the set { x: 4>(x) c U } (resp. { x: 4 (x) U%A } ) is open in X for each open set U in Y. Note that if f is an u.s.c. (l.s.c.) real-valued function, then 4X, defined by 1(x) = { r: r attains a maximum (minimum) if the family {I (x): x EX} has a maximal (minimal) member with respect to set inclusion. In the sequel X and Y are always T1 topological spaces, and 2y is the set of all nonempty closed subsets of Y. If a is any infinite cardinal, then we say that X is a-compact if each open cover of X having cardinality
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As with metric spaces, every uniform space X has a Hausdorff completion: that is, there exists a complete Hausdorff uniform space Y and a uniformly continuous map i: X → Y with the following property:
for any uniformly continuous mapping f of X into a complete Hausdorff uniform space Z, there is a unique uniformly continuous map g: Y → Z such that f = gi.
The Hausdorff completion Y is unique up to isomorphism. As a set, Y can be taken to consist of the minimal Cauchy filters on X. As the neighbourhood filter B(x) of each point x in X is a minimal Cauchy filter, the map i can be defined by mapping x to B(x). The map i thus defined is in general not injective; in fact, the graph of the equivalence relation i(x) = i(x ') is the intersection of all entourages of X, and thus i is injective precisely when X is Hausdorff.
The uniform structure on Y is defined as follows: for each symmetric entourage V (i.e., such that (x,y) is in V precisely when (y,x) is in V), let C(V) be the set of all pairs (F,G) of minimal Cauchy filters which have in common at least one V-small set. The sets C(V) can be shown to form a fundamental system of entourages; Y is equipped with the uniform structure thus defined.
The set i(X) is then a dense subset of Y. If X is Hausdorff, then i is an isomorphism onto i(X), and thus X can be identified with a dense subset of its completion. Moreover, i(X) is always Hausdorff; it is called the Hausdorff uniform space associated with X. If R denotes the equivalence relation i(x) = i(x '), then the quotient space X/R is homeomorphic to i(X).
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It is shown here how G. D. Birkhoff's notion of the center of a homeomorphism or flow naturally gives rise to an analytic set in a product space. It is shown that for a wide class of spaces this set is not a Borel set. Let X be a locally compact separable metric space with complete metric d and let H(X) be the space of autohomeomorphisms of X. The space H(X) has a topology under which it is a complete separable metric group [6, 9]. For a wide class of Xs, it is known that this topology is unique [7]. This topology may be briefly described as follows. Let X* = X U {oo} be the one point compactification of X and consider the space M = M(X*, X*) of all continuous maps of X* into X* provided with the compact open topology [9]. In this topology, M is a Polish space: M is separable and possesses a complete metric compatible with this topology. Identify H(X) with F = {(f, g) E M x M:fg = gf = idx* and f(oo) = oo}. Since F is closed in M x M, F is also a Polish space. We consider H(X) to have this topology. If h E H(X) and Y is an h-invariant subset of X, then a point y E Y is said to be nonwandering with respect to Y provided there is an increasing sequence of positive integers n1, n2, n3, . . . and points yp E Y, p = 1, 2, 3, . . . such that the sequence hnp(yp) converges to y. Let Rh(Y) = {y E Y:y is nonwandering with respect to Y}. If Y is a closed h-invariant set, then Rh(Y) is also closed and h-invariant. Set R(X) = X and by recursion, for each ordinal ox, Ro '(X) = Rh(Ro'(X)) and, if X is a limit ordinal, Rx(X) = n,a
Center (category theory)
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The notion of base-mesocompact mapping is introduced and the following results are proved:①Let f:X→Y be a closed Lindeloff mapping.If X is regular,then f:X→Y is base-countable mesocompact mapping;② Let X and Y be both base-mesocompact.X×Y is base-countable mesocompact if Y is locally compact.
Base (topology)
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the importance of the following Problem. Let A : X—*Y be a mapping (not necessarily linear) of a topological space X into a topological space Y. Under what conditions is A (X) open in F? The aim of this paper is to give a particular solution of this problem in the case of mappings A : X—>X of a Banach space X into itself. It will be shown that the Fixed Point Theorems of Schauder and Brouwer may be applied to find conditions under which the image A(X) of X is open in X. The idea of the following proofs is: Suppose that A: X-+X is a mapping of X into itself and let yoŒA(X). To prove that A (X) contains a spherical region S(ya, r
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