On rings of real valued clopen continuous functions

Among variant kinds of strong continuity in the literature, the clopen continuity or cl-supercontinuity (i.e., inverse image of every open set is a union of clopen sets) is considered in this paper.  We investigate and study the ring Cs(X) of all real valued clopen continuous functions on a topological space X.  It is shown that every ƒ ∈ Cs(X) is constant on each quasi-component in X and using this fact we show that Cs(X) ≅ C(Y), where Y is a zero-dimensional s-quotient space of X.  Whenever X is locally connected, we observe  that Cs(X) ≅ C(Y),  where Y is a discrete space.  Maximal ideals of Cs(X) are characterized in terms of quasi-components in X and it turns out that X  is mildly compact(every clopen cover has a finite subcover) if and only if every maximal ideal  of Cs(X)is  fixed. It is shown that the socle of Cs(X) is  an essential ideal if and only if the union of all open quasi-components in X is s-dense.  Finally the counterparts of some familiar spaces, such as Ps-spaces, almost Ps-spaces, s-basically and s-extremally disconnected spaces  are  defined  and  some  algebraic  characterizations  of  them  are given via the ring Cs(X)

On e-spaces and rings of real valued e-continuous functions

Whenever the closure of an open set is also open, it is called e-open and if a space have a base consisting of e-open sets, it is called e-space. In this paper we first introduce and study e-spaces and e-continuous functions (we call a function f from a space X to a space Y an e-continuous at x ∈ X if for each open set V containing f(x) there is an e-open set containing x with f ( U ) ⊆ V ). We observe that the quasicomponent of each point in a space X is determined by e-continuous functions on X and it is characterized as the largest set containing the point on which every e-continuous function on X is constant. Next, we study the rings Ce( X ) of all real valued e-continuous functions on a space X. It turns out that Ce( X ) coincides with the ring of real valued clopen continuous functions on X which is a C(Y) for a zero-dimensional space Y whose elements are the quasicomponents of X. Using this fact we characterize the real maximal ideals of Ce( X ) and also give a natural representation of its maximal ideals. Finally we have shown that Ce( X ) determines the topology of X if and only if it is a zero-dimensional space

Identifying the Components of a Spiritual Curriculum Using Interpretive Structural Modeling (ISM)

The present study identifies and determines the components. The spiritual curriculum uses interpretive structural modeling. The research method was applied research in terms of orientation, descriptive approach in terms of deduction, descriptive purpose in terms of research philosophy and mixed (qualitative-quantitative) in terms of research philosophy. The statistical population of the study consisted of experts in the field of educational management. The sampling method was purposive and the number of participants based on theoretical saturation was equal to 14 people. Data were collected in the qualitative part using semi-structured interviews and in the quantitative part using a questionnaire whose validity and reliability were confirmed by experts. The results showed that the components of the spiritual curriculum include self-improvement, self-worth, sociology, sociability, globalization, globalization, spirituality, spiritualization, the need for evolution, the need for meaning, introspection and between is individual. The findings of this study also showed that the components of self-improvement, self-worth and the need for further development are more effective than the components of globalization, sociology, spiritualization, intrapersonal, interpersonal and more sociability