B-3 - Product Spaces

A categorical characterization of a subcategory S of TOP (or 7J) is one which enables the identification of S in TOP (or 7^) without requiring the reconstruction of the topological structure of its objects.In this paper we so characterize various familiar subcategories of TOP (Hausdorff spaces, normal spaces, compact Hausdorff spaces, paracompact Hausdorff spaces, metrizable spaces, first countable spaces) in terms of the global behavior of the (objects and) morphisms of the subcategory. Introduction. A categorical characterization of a subcategory S of TOP (or of T2) is one which enables the identification of S in TOP (or T2) without requiring the reconstruction of the topological structure of its objects.The proof of the triviality of the automorphism class group of TOP [Fe, p. 32] assures us of the existence of such characterizations.As will be shown in §1, this proof establishes that all the topological structure of an object of TOP can be recovered from the behavior of the arrows with that object as source and/ or target.In §2 we establish a number of lemmas (mostly known), and we use these lemmas in §3 to characterize many important subcategories of TOP and/ or T2, such as the metrizable spaces, the paracompact Hausdorff spaces, the normal spaces, etc.We can distinguish at least three classes of "categorical" characterizations of subcategories of TOP, which are listed below: Class 3. Straightforward translations of the familiar definitions.These can always be obtained because of the results in §1.Example: Characterize the concepts "open", "cover", "refinement", "locally finite" categorically and then the standard definition of paracompact spaces is categorical.Class 2. Characterizations using standard categorical constructions (product, coproduct, epimorphic images, etc.) but in terms of individual objects, and/or constants outside the subcategory.Example: The category of paracompact Hausdorff spaces contains precisely those Hausdorff spaces X such that

We remind two facts for topological spaces. The one is that in a Hausdorff space $X$ each sequence has a unique limit. This allows us to have a function from the set of all sequences in $X$ to $X$. Another is that in the first countable spaces some topological objects such as open subsets, closed subsets, closures and interiors of the sets, continuous functions and many others can be defined in terms of convergent sequences. In this paper we compare these notions with their sequential versions in topological spaces. We will take the product spaces into account and give some results.

Čech systems that induce approximate inverse systems

Totally Lindelöf and totally ω-narrow paratopological groups

b-13 - Special Spaces

We develop a new route through which to explore $\ker \Psi_{X}$, the kernel of the $\pi_{1}$-shape group homomorphism determined by a general space $X$, and establish, for each locally path connected, paracompact Hausdorff space $% X$, $\ker\Psi_{X}$ is precisely the Spanier group of $X$.

In this paper, the concepts of $\omega$-open and $\omega$-closed functions between topological spaces will be introduced and characterized. Moreover, related concepts such as $\omega$-connectedness and $\omega$-continuity from an arbitarary topological space into the product space will also be characterized.

The central notion in topology is the concept of “continuity of functions” between topological spaces. A discussion of this concept and some other notions related to mappings is the object of the first section. In the second section, we consider the problem of topologizing the Cartesian products of a family of topological spaces in some natural and useful way. Here, we are mainly concerned with the “Tychonoff topology,” which is the smallest topology for a product of topological spaces such that each projection is continuous.

The lexicographic ordered products and the usual Tychonoff products

We develop a new epi-convergence based on the use of bounded convergent nets on the product topology of the strong topology on the primal space and weak star topology on the dual space of a general real Banach space. We study the propagation of the associated variational convergences through conjugation of convex functions defined on this product space. These results are then applied to the problem of construction of a bigger-conjugate representative function for the recession operator associated with a maximal monotone operator on this real Banach space. This is then used to study the relationship between the recession operator of a maximal monotone operator and the normal–cone operator associated with the closed, convex hull of the domain of that monotone operator. This allows us to show that the strong closure of the domain of any maximal monotone operator is convex in a general real Banach space.

A topological space S P , which is a modification of the Sorgenfrey line S, is considered. It is defined as follows: if x ∈ P ⊂ S, then a base of neighborhoods of x is the family {[x, x + e), e > 0} of half-open intervals, and if x ∈ SP, then a base of neighborhoods of x is the family {(x − e, x], e > 0}. A necessary and sufficient condition under which the space S P is homeomorphic to S is obtained. Similar questions were considered by V. A. Chatyrko and I. Hattori, who defined the neighborhoods of x ∈ P to be the same as in the natural topology of the real line.

D. Edmundson [1] has recently constructed a counter-example to the conjecture by A. D. Wallace that every compact connected topological lattice is distributive. He discovered a subset of Euclidean three-space which, with the appropriate lattice operations, forms a nonmodular compact connected topological lattice. The main result of this paper states that if a locally compact connected subset of Euclidean two-space admits a pair of continuous lattice operations then the lattice is distributive. 1. Preliminaries. We recall that a topological lattice is a Hausdorff space, L, together with a pair of continuous furnctions A: L XL-*yL and V: L XL- yL which satisfy the usual conditions stipulated for a lattice. As is usual we denote A(x, y) byxAy and V(x, y) byxVy. Unless explicitly stated to the contrary, we reserve the symbols 0 and 1 to denote the unique minimal and maximal elements of a lattice, whenever they exist. A subset, C, of L is a chain if xAy=x or y for any pair of elements x and y in C. If a and b are elements of L with a ? b then C is said to be a chain from a to b provided C is a chain contained in a V(bAL) and containing both a and b. If A is a subset of L, we deiiote by A*, A' and F(A) =A *\A' the closure, interior and boundary of A respectively. Throughout this paper, R2 will denote the Cartesian plane with the usual topology, SI will deniote the unit circle and I will denote the closed real number interval [(, 1]. A subset of R2 is a simple closed curve if it is a homeomorph of S1. We reccll that Jordan's theorem [2] states that any simple closed curve cuts R2 into exactly two components, one bounded and the other unbounded. If C is a simple closed curve, we denote by B(C) the bounded component of R2\C. If X is a topological space and A is a subset of X, we denote by IIn(X, A) the n-dimensional Alexander-Kolmogoroff cohomology group of X modulo A with coefficients in some fixed nontrivial additive abelian group. In this paper we utilize two dimension functions, namely codimension (cd) and inductive dimension (ind). For the definitions and essential theorems relating to these dimension functions the reader is referred to the work of H. Cohen [3]. It is a pleasure to acknowledge the advice and suggestions of A. D. Wallace in the preparation of this paper.

This paper is concerned with topological spaces whose continuous maps into a given space R are constant, as well as with spaces having this property locally. We call these spaces R-monolithic and locally R-monolithic, respectively.Spaces with such properties have been considered in [1], [5]-[7], [10], [11], [22], [28], [31], where with the exception of [10], the given space R is the set of real-numbers with the usual topology. Obviously, for a countable space, connectedness is equivalent to the property that every continuous real-valued map is constant. Countable connected (locally connected) spaces have been constructed in papers [2]-[4], [8], [9], [11]-[21], [23]-[26], [30].

This chapter discusses topological spaces and associated concepts, including first‐ and second‐countability, compactness, and separation properties. Weak topologies are defined. The case of product spaces and the product topology are treated, the Tychonoff theorem, and ideas of embedding, compactification, and metrization.

A n-dimensional topological manifold is defined as an n-dimensional local Euclidean space M that is also a second countable space and a Hausdorff space. If a topological space has a countable base, it is referred to as a second countable space. At the same time, continuous function and metric space can help people to understand topological space better. This article will express some basic ideas about continuous functions, metric space, topological space and its properties, and topological manifolds. Moreover, the paper shows some basic ideas of how topological manifolds, topological space and function could be recognized and proved. Through analysis, this paper demonstrates the connection between them, such as using properties of continuous function to prove the definition and properties of topological manifolds and spaces.