ABSTRACT

Let *X be a nonstandard model of the topological space (X, τ). Fundamental in nonstandard topology is the notion of the monad of a point x ∈ X which is by definition the set m ( x )     : = ∩ U ∈ τ ,   x ∈ U * U https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780367811631/7876cf56-2aaa-47b0-892b-62310007e556/content/inq_chapter4_42_1.tif"/> . Instead of y ∈ m(x) we use as well the more intuitive notion y ≈ x. The set n s   * X     : = ∪ x ∈ X m ( x ) https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780367811631/7876cf56-2aaa-47b0-892b-62310007e556/content/inq_chapter4_42_2.tif"/> is called the set of all nearstandard points. If A is a subset of the topological space (X, τ) the monad of A is the set m τ ( A )     : = ∩ U ∈ τ ,   A ⊂ U   * U https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780367811631/7876cf56-2aaa-47b0-892b-62310007e556/content/inq_chapter4_42_3.tif"/> If K is a compact subset of X then the relation m τ ( K )   = ∪ x ∈ K m ( x ) https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780367811631/7876cf56-2aaa-47b0-892b-62310007e556/content/inq_chapter4_42_4.tif"/> holds. If ℱ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780367811631/7876cf56-2aaa-47b0-892b-62310007e556/content/inq_chapter4_42_5.tif"/> is a filter on X the monad of the filter ℱ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780367811631/7876cf56-2aaa-47b0-892b-62310007e556/content/inq_chapter4_42_6.tif"/> is the set m ( ℱ )     : = ∩ F ∈ ℱ   * F https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780367811631/7876cf56-2aaa-47b0-892b-62310007e556/content/inq_chapter4_42_7.tif"/> . For a general treatment of monads we refer to [4]. Let k be the system of all compact subsets of the topological space X. Then the set c p t * X     : = ∪ K ∈ k * K https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780367811631/7876cf56-2aaa-47b0-892b-62310007e556/content/inq_chapter4_42_8.tif"/> is called the set of all compact points. It is well known that the equality ns*X = cpt*X means that X is locally compact, i.e., that every point possesses a compact neighborhood.