Let X be a topological space and A⊆X. x∊X is an isolated points of A if There exist the open set in X such that G∩A={x}.
Especially if A=X then x∊X be an isolated points if and only if {x} be an Open.
In this research we study the space of spec(R) and zariski topology that Display is by (spec(R) , τ_z).
We remember that in zariski topology the prime ideal P is an isolated Points if and only if be prime minimal. Also the topological space is Scattered space if every subspace of that have one isolated points.
In this research we study the isolated points in arbitrary topological Space (X , τ).
If the τ- specialization pre- order on X has enough maximal elements , Then a point x∊X is an isolated point in (X , τ) if and only if x is both an Isolated point in the subspace of τ-kerneled points of X and in the τ-Closure of {x}. this result is applied to an arbitrary subspace of the prime Spectrum spec(R) of a ( commutative with nonzero identity ) ring R , and In particular , to the space spec(R) and the maximal and minimal spectrum Of R. finally we characterize when spec(R) is scattered space.
 

In this thesis we present a description for generalizd closur and isotonic space.
Then we identify their elementary characterisation by defenition of continious function
in these spaces. finally we investigate connected, Z-connected and strongly connected
spaces.

 

 

We present a new generalization of realcompactness called r-realcompact
space that based on ultrafilters of regular -subsets.we also introduce some
of the properties of the r-realcompact space.Finally we investigate its relationship
with realcompactness,Almost realcompactness,Almost*realcompactness,
C-realcompact.

 

 In this thesis, a subring of the ring of continuous functions on a
topological space 𝑌 with values in a linearly ordered field 𝐺 equipped with its
order topology, namely the ring of continuous functions with compact support
is investigated. Unless 𝑌 is compact, these rings are commutative rings without
unity. However, unlike many other commutative rings without unity, these
rings turn out to have some nice properties, essentially in determining the
property of 𝑌 being locally compact non-compact or the property of 𝑌 being
nowhere locally compact. Also, one can associate with these rings a topological
space resembling the structure space of a commutative ring with unity, such
that the classical Banach Stone Theorem can be generalized to the case when
the range field is that of the reals.

The aim of this study is to investigate the background of logarithm emergence and methods of its calculation, from the past to present periods and how the laws governing it .In first chapter, It has been investigated the definition of logarithm and its theorems and also a glimpse into the history of it. In the second chapter, it has been discussed to prove that Napier number is irrational. In the third chapter, it also has been discussed the tables of logarithms by Edward Sang and it has been concisely brought the names of mathematicians who worked on the tables of logarithms and consequently about the methods for calculating logarithms of numbers. In the second part of the third chapter, the expansion of the logarithm in Spain in the 17th and 18th centuries and its impact on the sciences of that time and the names of books and the scientists who have worked in the field of logarithms were discussed. In the fourth chapter, It was discussed the use of logarithms in the various fields including calculation of earthquake intensity, voice, chemistry, music as well as the use of Napier number about the growth and deterioration on topics such as forecasting population growth of bacteria and finally a lot of subjects about the use of logarithms in the science of music were discussed .Finally, in the fifth and final chapter on the impact of computers in the mathematics education were discussed and also it was discussed about teaching logarithms by software such as Flash and GeoGebra to help students have a better understanding of the properties of logarithm based on their visualizations.

 This study aims to explore the concepts of infinity, infinitesimal, infinite series, also the use of different approaches of scientists using the same concepts within the history with the purpose of improving the training methods. This study is kind of descriptive and library one. It has been triede to provide a historical evalution to discover the concepts of infinity, infinitesimal, infinite series, and to present the applied methods by mathematician dealing with these concepts within the history. This is a good sample to demonstrate that by the use of historical researches on a concept, simple methods can be found to train the students. As a result, the importance of the study confirms the history of mathematics.