scholarly journals Maximal perfect spaces

1972 ◽  
Vol 7 (3) ◽  
pp. 429-436
Author(s):  
Ivan Baggs
Keyword(s):  
Topological Space ◽  
Limit Point ◽  
Hausdorff Space ◽  
Hausdorff Spaces

Let (X, T) be a topological space (we assume T1. throughout) where every point is a limit point. The purpose of this note is to present an internal construction of a maximal perfect topology on (X, T). The existence of a maximal connected Hausdorff space has not been demonstrated. However, this construction of a maximal perfect topology is useful in constructing connected Hausdorff spaces which cannot be embedded in a maximal connected Hausdorff space.

scholarly journals Compact and extremally disconnected spaces

2004 ◽  
Vol 2004 (20) ◽  
pp. 1047-1056
Author(s):  
Bhamini M. P. Nayar
Keyword(s):  
Topological Space ◽  
Compact Space ◽  
Closed Subset ◽  
Hausdorff Space ◽  
Hausdorff Spaces ◽  
Compact Spaces ◽  
Hausdorff Topological Space ◽  
Extremally Disconnected

Viglino defined a Hausdorff topological space to beC-compact if each closed subset of the space is anH-set in the sense of Veličko. In this paper, we study the class of Hausdorff spaces characterized by the property that each closed subset is anS-set in the sense of Dickman and Krystock. Such spaces are calledC-s-compact. Recently, the notion of strongly subclosed relation, introduced by Joseph, has been utilized to characterizeC-compact spaces as those with the property that each function from the space to a Hausdorff space with a strongly subclosed inverse is closed. Here, it is shown thatC-s-compact spaces are characterized by the property that each function from the space to a Hausdorff space with a strongly sub-semiclosed inverse is a closed function. It is established that this class of spaces is the same as the class of Hausdorff, compact, and extremally disconnected spaces. The class ofC-s-compact spaces is properly contained in the class ofC-compact spaces as well as in the class ofS-closed spaces of Thompson. In general, a compact space need not beC-s-compact. The product of twoC-s-compact spaces need not beC-s-compact.

Epimorphisms From S(X) onto S(Y)

1986 ◽  
Vol 38 (3) ◽  
pp. 538-551 ◽  
Author(s):  
K. D. Magill ◽  
P. R. Misra ◽  
U. B. Tewari
Keyword(s):  
Topological Space ◽  
Hausdorff Spaces ◽  
Severe Restriction ◽  
Completely Regular ◽  
Image Position

1. Introduction. In this paper, the expression topological space will always mean generated space, that is any T1 space X for whichforms a subbasis for the closed subsets of X. This is not at all a severe restriction since generated spaces include all completely regular Hausdorff spaces which contain an arc as well as all 0-dimensional Hausdorff spaces [3, pp. 198-201], [4].The symbol S(X) denotes the semigroup, under composition, of all continuous selfmaps of the topological space X. This paper really grew out of our efforts to determine all those congruences σ on S(X) such that S(X)/σ is isomorphic to S(Y) for some space Y.

Projective Elements in Categories with Perfect θ-Continuous Maps

1981 ◽  
Vol 33 (4) ◽  
pp. 872-884 ◽  
Author(s):  
Hans Vermeer ◽  
Evert Wattel
Keyword(s):  
Hausdorff Space ◽  
Continuous Functions ◽  
List Type ◽  
Hausdorff Spaces ◽  
Extremally Disconnected ◽  
Continuous Maps

In 1958 Gleason [6] proved the following :THEOREM. In the category of compact Hausdorff spaces and continuous maps, the projective elements are precisely the extremally disconnected spaces.The projective elements in many topological categories with perfect continuous functions as morphisms have been found since that time. For example: In the following categories the projective elements are precisely the extremally disconnected spaces:(i) The category of Tychonov spaces and perfect continuous functions. [4] [11].(ii) The category of regular spaces and perfect continuous functions. [4] [12].(iii) The category of Hausdorff spaces and perfect continuous functions. [10] [1].(iv) In the category of Hausdorff spaces and continuous k-maps the projective members are precisely the extremally disconnected k-spaces. [14].In 1963 Iliadis [7] constructed for every Hausdorff space X the so called Iliadis absolute E[X], which is a maximal pre-image of X under irreducible θ-continuous maps.

Spaces which Cannot be Written as a Countable Disjoint Union of Closed Subsets

1973 ◽  
Vol 16 (3) ◽  
pp. 435-437 ◽  
Author(s):  
C. Eberhart ◽  
J. B. Fugate ◽  
L. Mohler
Keyword(s):  
Hausdorff Space ◽  
Disjoint Union ◽  
Hausdorff Spaces ◽  
Locally Compact ◽  
The One ◽  
One Point Compactification ◽  
Compact Subsets

It is well known (see [3](1)) that no continuum (i.e. compact, connected, Hausdorff space) can be written as a countable disjoint union of its (nonvoid) closed subsets. This result can be generalized in two ways into the setting of locally compact, connected, Hausdorff spaces. Using the one point compactification of a locally compact, connected, Hausdorff space X one can easily show that X cannot be written as a countable disjoint union of compact subsets. If one makes the further assumption that X is locally connected, then one can show that X cannot be written as a countable disjoint union of closed subsets.(2)

A Topological Banach Fixed Point Theorem for Compact Hausdorff Spaces

1994 ◽  
Vol 37 (4) ◽  
pp. 552-555 ◽  
Author(s):  
Juris Steprans ◽  
Stephen Watson ◽  
Winfried Just
Keyword(s):  
Fixed Point ◽  
Hausdorff Space ◽  
Unique Fixed Point ◽  
Banach Fixed Point Theorem ◽  
Hausdorff Spaces ◽  
Compact Metric Space ◽  
Banach Contraction ◽  
Compact Metrizable Space ◽  
Admissible Metric ◽  
The Banach Contraction Principle

AbstractWe propose an analogue of the Banach contraction principle for connected compact Hausdorff spaces. We define a J-contraction of a connected compact Hausdorff space. We show that every contraction of a compact metric space is a J-contraction and that any J-contraction of a compact metrizable space is a contraction for some admissible metric. We show that every J-contraction has a unique fixed point and that the orbit of each point converges to this fixed point.

Simple Links in Locally Compact Connected Hausdorff Spaces are Nondegenerate

1981 ◽  
Vol 34 (2) ◽  
pp. 349-355
Author(s):  
David John
Keyword(s):  
Hausdorff Space ◽  
Metric Spaces ◽  
Connected Space ◽  
Hausdorff Spaces ◽  
Locally Compact ◽  
Peano Continua

The fact that simple links in locally compact connected metric spaces are nondegenerate was probably first established by C. Kuratowski and G. T. Whyburn in [2], where it is proved for Peano continua. J. L. Kelley in [3] established it for arbitrary metric continua, and A. D. Wallace extended the theorem to Hausdorff continua in [4]. In [1], B. Lehman proved this theorem for locally compact, locally connected Hausdorff spaces. We will show that the locally connected property is not necessary.A continuum is a compact connected Hausdorff space. For any two points a and b of a connected space M, E(a, b) denotes the set of all points of M which separate a from b in M. The interval ab of M is the set E(a, b) ∪ {a, b}.

Zero-Dimensional Compactifications of Locally Compact Spaces

1974 ◽  
Vol 26 (4) ◽  
pp. 920-930 ◽  
Author(s):  
R. Grant Woods
Keyword(s):  
Topological Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Dense Subspace ◽  
Locally Compact ◽  
Partially Order ◽  
Compact Spaces ◽  
Hausdorff Topological Space ◽  
Continuous Map ◽  
Locally Compact Spaces

Let X be a locally compact Hausdorff topological space. A compactification of X is a compact Hausdorff space which contains X as a dense subspace. Two compactifications αX and γX of X are equivalent if there is a homeomorphism from αX onto γX that fixes X pointwise. We shall identify equivalent compactifications of a given space. If is a family of compactifications of X, we can partially order by saying that αX ≦ γX if there is a continuous map from γX onto αX that fixes X pointwise.

Local Topological Properties of Maps and Open Extensions of Maps

1977 ◽  
Vol 29 (6) ◽  
pp. 1121-1128
Author(s):  
J. K. Kohli
Keyword(s):  
Topological Space ◽  
Open Subset ◽  
Countable Union ◽  
Dense Open Subset ◽  
Topological Properties ◽  
Hausdorff Spaces ◽  
Local Homeomorphism ◽  
Open Map

A σ-discrete set in a topological space is a set which is a countable union of discrete closed subsets. A mapping ƒ : X ⟶ Y from a topological space X into a topological space Y is said to be σ-discrete (countable) if each fibre ƒ-1(y), y ϵ Y is σ-discrete (countable). In 1936, Alexandroff showed that every open map of a bounded multiplicity between Hausdorff spaces is a local homeomorphism on a dense open subset of the domain [2].

scholarly journals On projective lift and orbit spaces

1994 ◽  
Vol 50 (3) ◽  
pp. 445-449 ◽  
Author(s):  
T.K. Das
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Covariant Functor ◽  
Hausdorff Spaces ◽  
Orbit Spaces ◽  
Locally Compact ◽  
Discontinuous Group ◽  
Coreflective Subcategory ◽  
Group Of Homeomorphisms ◽  
Locally Compact Hausdorff Space

By constructing the projective lift of a dp-epimorphism, we find a covariant functor E from the category Cd of regular Hausdorff spaces and continuous dp-epimorphisms to its coreflective subcategory εd consisting of projective objects of Cd We use E to show that E(X/G) is homeomorphic to EX/G whenever G is a properly discontinuous group of homeomorphisms of a locally compact Hausdorff space X and X/G is an object of Cd.

A note on λ-compact spaces

2013 ◽  
Vol 63 (6) ◽  
Author(s):  
O. Karamzadeh ◽  
M. Namdari ◽  
M. Siavoshi
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Hausdorff Spaces ◽  
Compact Spaces

AbstractWe extend the well-known and important fact that “a topological space X is compact if and only if every ideal in C(X) is fixed”, to more general topological spaces. Some interesting consequences are also observed. In particular, the maximality of compact Hausdorff spaces with respect to the property of compactness is generalized and the topological spaces with this generalized property are characterized.

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