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On the largest coreflective Cartesian closed subconstruct of Prtop

E. Lowen-Colebunders, G. Sonck
1996 Applied Categorical Structures  
This implies that in any coreflective subconstruct of Prtop, exponential objects are finitely generated.  ...  Moreover, in any finitely productive, coreflective subconstruct, exponential objects are precisely those objects of the subconstruct that are finitely generated.  ...  Since we work only with well-fibred topological constructs C, we use the following characterization of an exponential object: X is exponential in C if and only if the functor X × -preserves final epi-sinks  ... 
doi:10.1007/bf00124115 fatcat:nrhitewlovcgdady2wuxxg3soy

On the Largest Coreflective Cartesian Closed Subconstruct of Prtop [chapter]

E. Lowen-Colebunders, G. Sonck
1996 Categorical Topology  
This implies that in any coreflective subconstruct of Prtop, exponential objects are finitely generated.  ...  Moreover, in any finitely productive, coreflective subconstruct, exponential objects are precisely those objects of the subconstruct that are finitely generated.  ...  Since we work only with well-fibred topological constructs C, we use the following characterization of an exponential object: X is exponential in C if and only if the functor X × -preserves final epi-sinks  ... 
doi:10.1007/978-94-009-0263-3_6 fatcat:wrrm6gpssbdyrcplweifgulcom

Exponential objects and Cartesian closedness in the constructPrtop

E. Lowen-Colebunders, G. Sonck
1993 Applied Categorical Structures  
We give an internal characterization of the exponential objects in the construct Prtop and investigate Cartesian closedness for coreflective or topological full subconstructs of Prtop.  ...  With regard to topological full subconstructs of Prtop we give an example of a Cartesian closed one that is large enough to contain all topological Frtchet spaces and all TI pretopological Frtchet spaces  ...  This result disproves the conjecture formulated in [16] that the exponential objects in Prtop can be characterized by some filter-theoretic description of core-compactness.  ... 
doi:10.1007/bf00872940 fatcat:a4hcthtsabcntfdjzzbojdxptq

Classes of pretopological spaces closed under the formation of final structures

E. Lowen-Colebunders, G. Sonck
1996 Topology and its Applications  
We investigate coreflective subconstructs of the construct h-top of pretopological spaces and continuous maps and in particular the inclusion "order" between these subconstructs.  ...  Using these minimal elements we obtain a "partition" of the whole conglomerate of coreflective subconstructs of &top.  ...  In the topological construct Prtop of pretopological spaces and continuous maps final structures are formed in a very easy and elegant way. Moreover quotients in Prtop are hereditary [3, 12] .  ... 
doi:10.1016/0166-8641(95)00097-6 fatcat:tfcpep7kkfavjj4gmmw6pha2sa

Page 7272 of Mathematical Reviews Vol. , Issue 94m [page]

1994 Mathematical Reviews  
Summary: “We give an internal characterization of the exponential objects in the construct Prtop and investigate Cartesian closedness for co-reflective or topological full subconstructs of Prtop.  ...  (B-VUB; Brussels) Exponential objects and Cartesian closedness in the construct Prtop. (English summary) Appl. Categ. Structures 1 (1993), no. 4, 345-360.  ... 

Page 1746 of Mathematical Reviews Vol. , Issue 97C [page]

1997 Mathematical Reviews  
Summary: “We give characterizations of perfect maps (as well as of continuous maps and compact maps) in metric spaces in terms  ...  This implies that in any coreflective subconstruct of Prtop, exponential ob- jects are finitely generated.  ... 

Injective Convergence Spaces and Equilogical Spaces via Pretopological Spaces

Reinhold Heckmann
2006 Electronical Notes in Theoretical Computer Science  
On the other hand, we study the category PrTop of pretopological spaces that lies in-between Top and Conv/Equ, identify its injective spaces, and show that they are also injective in Conv and Equ.  ...  Sierpinski space Ω is injective in the category Top of topological spaces, but not in any of the larger cartesian closed categories Conv of convergence spaces and Equ of equilogical spaces.  ...  Such a characterization is also possible in case of PrTop, but is actually much simpler.  ... 
doi:10.1016/j.entcs.2005.11.065 fatcat:q2entdrus5ccvkgethlw5ccqea

Exponentiation for unitary structures

Dirk Hofmann
2006 Topology and its Applications  
For T = U the ultrafilter monad, we characterize exponentiable morphisms in Alg u (U; V).  ...  Further, we give a sufficient condition for an object to be exponentiable in the category Alg(U; V) of reflexive and transitive lax algebras.  ...  Our result covers the characterization of exponentiable morphisms in PrTop obtained in [24] , of exponentiable objects in PrAp obtained in [20] , and it gives a new characterization of exponentiable  ... 
doi:10.1016/j.topol.2005.01.037 fatcat:zjjxs34vdfbepjiln4oxjt5s44

Topological improvements of categories of structured sets

Horst Herrlich
1987 Topology and its Applications  
Hulls In this section various extensions of a construct A, i.e., full concrete embeddings E : A+ P(A) are characterized.  ...  Of the familiar topological constructs, listed in 1.1, only PrTop, PsTop, Conv, Mer, Bor, Simp and Rere are hereditary.  ...  Reiterman, The quasitopos hull of the category of uniform spaces, Topology Appl. 27 (1987)  ... 
doi:10.1016/0166-8641(87)90101-5 fatcat:swfvqfnmabcmrgftsncjclvzhu

Page 3677 of Mathematical Reviews Vol. , Issue 2004e [page]

2004 Mathematical Reviews  
However it is shown that the three generalized functors are equiv- alent.” 2004e:18005 18B30 18A20 54B30 54C10 Richter, Giinther (D-BLFM; Bielefeld) A characterization of exponentiable maps in PrTop.  ...  Surprisingly or not, it turns out to characterize exponentiable maps.”  ... 

Initially dense objects for metrically generated theories

Veerle Claes
2009 Topology and its Applications  
a r t i c l e i n f o a b s t r a c t MSC: 54B30 54A05 18B99 We study initially dense objects for metrically generated constructs X in the sense of [E. Colebunders, R.  ...  For the base categories consisting of metrics, quasi-metrics, totally bounded quasi-metrics and totally bounded metrics, a general description of some initially dense objects is given in case the expander  ...  Acknowledgement I thank Eva Vandersmissen for her valuable comments towards some examples which appear in this paper.  ... 
doi:10.1016/j.topol.2009.03.023 fatcat:b5fftbs66ra3jk6boslxs5ykta

The quasitopos hull of the construct of closure spaces

Veerle Claes, G. Sonck
2003 Applied General Topology  
The topological construct Cls of closure spaces and continuous maps is not a quasitopos. In this article we give an explicit description of the quasitopos topological hull of Cls using a method of F.  ...  <p>In the list of convenience properties for topological constructs the property of being a quasitopos is one of the most interesting ones for investigations in function spaces, differential calculus,  ...  In a well-fibred topological construct D, this notion can be characterized as follows: X is exponential in D iff for each D-object Y the set Hom D (X, Y ) can be supplied with the structure of a D-object  ... 
doi:10.4995/agt.2003.2006 fatcat:7lzxrebm2jf4zdhneucut2fgl4

Page 1474 of Mathematical Reviews Vol. , Issue Index [page]

Mathematical Reviews  
(English summary) 2003d:18002 — Non-symmetric exponential laws in the construct PrTop. (English summary) 2003f: 18008 Slapal, Josef Compactness with respect to a convergence structure.  ...  On t-pseudocompact mappings. (Russian. Russian summary) 20034:54023 Nordo, Giorgio (with Pasynkov, Boris A.) Characterizing continuity in topological spaces.  ... 

Page 317 of Mathematical Reviews Vol. 25, Issue Index [page]

Mathematical Reviews  
(English summary) [Euler-Jordan and Gauss functions and exponential in Burnside semirings}] 93f:18008 Kennison, John F.  ...  ., 93f:18007 Vazquez, Roberto Simple objects in categories. (Spanish) 93h:18007 Wang, Hao? (with Hu, Qing Ping) Four types of morphism-map categories and their properties. (Chinese.  ... 

Page 2094 of Mathematical Reviews Vol. , Issue Index [page]

Mathematical Reviews  
No. 150 (2003), 338-342. 81Q50 (81-05) Richter, Giinther' A characterization of exponentiable maps in PrTop. (English summary) Appl. Categ. Structures 11 (2003), no. 3, 261-265.  ...  (Summary) 2004e:18005 18B30 (18A20, 54B30, 54C10) — Exponentiability for maps means fibrewise core-compactness. (English summary) J. Pure Appl. Algebra 187 (2004), no. 1-3, 295-303. (D.  ... 
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