• The Mathematical Education
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• v.16 no.2
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• pp.22-26
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• 1978

It is shown that a map having an extension to an open map between the Alex-androff base compactifications of its domain and range has a unique such extension. J.S. Wasileski has introduced the Alexandroff base compactifications of Hausdorff spaces endowed with Alexandroff bases. We introduce a definition of morphism between such spaces to obtain a category which we denote by ABC. We prove that the Alexandroff base compactification on objects can be extended to a functor on ABC and that the compact objects give an epireflective subcategory of ABC. For each topological space X there exists a completely regular space $\alpha$X and a surjective continuous function $\alpha$$_{x}$ : Xlongrightarrow$\alpha$X such that for each completely regular space Z and g$\in$C (X, Z) there exists a unique g$\in$C($\alpha$X, 2) with g=g$^{\circ}$$\beta$$_{x}$. Such a pair ($\alpha$$_{x}$, $\alpha$X) is called a completely regularization of X. Let TOP be the category of topological spaces and continuous functions and let CREG be the category of completely regular spaces and continuous functions. The functor $\alpha$ : TOPlongrightarrowCREG is a completely regular reflection functor. For each topological space X there exists a compact Hausdorff space $\beta$X and a dense continuous function $\beta$x : Xlongrightarrow$\beta$X such that for each compact Hausdorff space K and g$\in$C (X, K) there exists a uniqueg$\in$C($\beta$X, K) with g=g$^{\circ}$$\beta$$_{x}$. Such a pair ($\beta$$_{x}$, $\beta$X) is called a Stone-Cech compactification of X. Let COMPT$_2$ be the category of compact Hausdorff spaces and continuous functions. The functor $\beta$ : TOPlongrightarrowCOMPT$_2$ is a compact reflection functor. For each topological space X there exists a realcompact space (equation omitted) and a dense continuous function (equation omitted) such that for each realcompact space Z and g$\in$C(X, 2) there exists a unique g$\in$C (equation omitted) with g=g$^{\circ}$(equation omitted). Such a pair (equation omitted) is called a Hewitt's realcompactification of X. Let RCOM be the category of realcompact spaces and continuous functions. The functor (equation omitted) : TOPlongrightarrowRCOM is a realcompact refection functor. In [2], D. Harris established the existence of a category of spaces and maps on which the Wallman compactification is an epirefiective functor. H. L. Bentley and S. A. Naimpally [1] generalized the result of Harris concerning the functorial properties of the Wallman compactification of a T$_1$-space. J. S. Wasileski [5] constructed a new compactification called Alexandroff base compactification. In order to fix our notations and for the sake of convenience. we begin with recalling reflection and Alexandroff base compactification.

• Biomedical Science Letters
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• v.13 no.4
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• pp.287-291
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• 2007

The factors and mechanisms by infection of Oriental Tsutsugamushi caused disease are not well understood. The onset of tsutsugamushi disease is characterized by chilliness, fever, malaise, headache and generalized aching. Infection of tsutsugamushi is the cause of impairment of function of a major organ often complicate the picture and immediately change the prognosis for the worse. Tsutsugamushi disease is reported that this disease is characterized by the histopathogenesis of liver, kidney, heart, and lung, but the variation of biochemical components in serum of tsutsugamushi disease patient are not clear. We analyzed total protein (TP), albumin (AL), aspartic aminotranferase (AST), alanine aminotransferase (ALT), alkaline phosphotase (ALP), urea nitrogen (UN), creatinine (CRE), glucose (GLD), cholesterol (CHOL) and total bilirubin (TB) in sera of patients with tsutsugamushi disease. In comparison with reference, total protein and albumin were abnormally decreased in 19.6% and 39.2% of patients, respectively. AST, ALT, ALP, creatinine, UN, glucose, cholesterol and total bilirubin were abnormally increased in 94.1 %, 72.5%, 25.5%, 15.7%, 9.8%, 62.7%, 25.5% and 6.0% of patients, respectively. The patients showed abnormal relative rate of protein electrophoretic fractions to total protein in serum compared to them of reference were 43.1% (albumin), 12.9% ($\alpha_1$-globulin), 58.8% ($\alpha_2$-globulin), 60.8% ($\beta$-globulin) and 70.6% ($\gamma$-globulin), respectively. These data suggest that infection of Oriental Tsutsugamushi causes impairment of function of a major organ and abnormal serum protein electrophoresis fractions to tsutsugamushi patients.