Alexandroff One Point Compactification

Abstract

Summary. In the article, I introduce the notions of the compactification of topological spaces and the Alexandroff one point compactification. Some properties of the locally compact spaces and one point compactification are proved. Let X be a topological space and let P be a family of subsets of X. We say that P is compact if and only if: (Def. 1) For every subset U of X such that U ∈ P holds U is compact.Let X be a topological space and let U be a subset of X. We say that U is relatively-compact if and only if: (Def. 2) U is compact.Let X be a topological space. Note that ∅ X is relatively-compact. Let X be a topological space. Observe that there exists a subset of X which is relatively-compact.Let X be a topological space and let U be a relatively-compact subset of X. Observe that U is compact.Let X be a topological space and let U be a subset of X. We introduce U is pre-compact as a synonym of U is relatively-compact.Let X be a non empty topological space. We introduce X is liminallycompact as a synonym of X is locally-compact.Let X be a non empty topological space. Let us observe that X is liminallycompact if and only if: (Def. 3) For every point x of X holds there exists a generalized basis of x which is compact.167

Double Series and Sums

“…( 23) ( κ α=0 R 1 (α)) κ∈N is convergent in the second coordinate if and only if the partial sums in the second coordinate of R 1 is convergent in the second coordinate. The theorem is a consequence of ( 19), (12), and (11). Let us consider a natural number k. Now we state the propositions:…”

“…) κ∈N is convergent in the first coordinate if and only if the partial sums in the first coordinate of R 1 is convergent in the first coordinate. The theorem is a consequence of ( 19), (12), and ( 11). ( 23) ( κ α=0 R 1 (α)) κ∈N is convergent in the second coordinate if and only if the partial sums in the second coordinate of R 1 is convergent in the second coordinate.…”

“…(iv) (the partial sums in the second coordinate of R 1 T ) T = the partial sums in the first coordinate of R 1 . The theorem is a consequence of (9). Let R 1 be a function from N × N into R. The functor ( κ α=0 R 1 (α)) κ∈N yielding a function from N × N into R is defined by the term (Def.…”

“…The notation and terminology used in this paper have been introduced in the following articles: [7], [1], [2], [18], [6], [9], [16], [11], [12], [23], [25], [30], [17], [3], [4], [13], [21], [20], [28], [29], [14], [22], [24], [27], and [15].…”

The Importance of Developing a Foundation for Naive Category Theory

“…1.1 that uses the notion of a set, such as given e.g. in [3]: the collections in the clauses (i) and (ii) of Def. 1.1 are then 'sets' in the sense of ZF.…”