المستخلص: |
In this thesis, we studied the weaker forms of Allexandroff spaces. Specifi cally, the g-Allexandroff, semi-Allexandroff, and pre-Allexandroff spaces. For g Allexandroff and semi-Allexandroff, we presented a survey study which includes a summary of all the definitions, similarities, differences, and everything related to these spaces. The other space pre-Allexandroff space, we introduced remarks and corrections for results in (Badry, 2013), and we get new results to complete the subject. We present new conclusions about pre-Allexandroff space with its proofs. We introduce and investigate a new weaker form of Allexandroff space which is called α-Allexandroff space at the same way which the previous weaker forms have been introduced. The characteristics are being created after careful study and analysis. The remarkable result achieved was direct sum of α-Alexadroff spaces is α-Alexadroff and converse is true. The other one is that the finite product of α-Alexadroff spaces is α-Alexadroff spaces but the converse need not be true.
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