On the cardinality of Hausdorff spaces and H-closed spaces

Abstract

ABSTRACT. We introduce the cardinal invariant aL ′ (X) and show that |X| ≤ 2 aL ′ (X)χ(X) for any Hausdorff space X (a corollary of Theorem 4.4). This invariant has the properties a) aL ′ (X) = ℵ0 if X is H-closed, and b) aL(X) ≤ aL ′ (X) ≤ aLc(X). Theorem 4.4 then gives a new improvement of the wellknown Hausdorff bound 2 L(X)χ(X) from which it follows that |X| ≤ 2 ψc (X) if X is H-closed (Dow/Porter [5]). The invariant aL ′ (X) is constructed using convergent open ultrafilters and an operator c : P(X) → P(X) with the property clA ⊆ c(A) ⊆ cl θ (A) for all A ⊆ X. As a comparison with this open ultrafilter approach, in §3 we additionally give a κ-filter variation of Hodel's proof [10] of the Dow-Porter result. Finally, for an infinite cardinal κ, in §5 we introduce κwH-closed spaces, κH ′ -closed spaces, and κH ′′ -closed spaces. The first two notions generalize the H-closed property. Key results in this connection are that a) if κ is an infinite cardinal and X a κwH-closed space with a dense set of isolated points such that χ(X) ≤ κ, then |X| ≤ 2 κ , and b) if X is κH ′ -closed or κH ′′ -closed then aL ′ (X) ≤ κ. This latter result relates these notions to the invariant aL ′ (X) and the operator c.

A useful common weakening of Lindelöf and H-closed

“…A positive answer to the weaker question concerning only the couple Lindelöf and H-closed is given in [2] and [4]. By combining the results from [2] and [3], in this short note we present a property P which provides a full positive answer to Question 1.…”

Research on the Adaptive Tracking Method for the Tracking of the Track of the Long-Jump Athletes

“…Wherein, r and θ are the coordinates of the frequency domain feature distribution of the long jump trajectory tracking image, and the dynamic characteristics of the long jump trajectory are analyzed in combination with the dynamic evolution feature distribution of the long jump trajectory [ 17 ]. The feature high resolution detection of the long jump track tracking image is carried out, and the detection function is obtained as follows: …”

On the cardinality of Urysohn spaces and weakly $H$-closed spaces

“…In this section we modify a filter construction given in [7]. In that paper an operatorĉ was constructed.…”

“…The analogue of the following proposition in the case of Hausdorff spaces is contained in the proof of Proposition 4.1 in [7].…”

“…Many researchers have worked on the cardinality of Urysohn and H-closed spaces, in particular considering cardinal invariants defined by using the θ-closure operator defined below (see, for instance, [1], [2], [4], [5], [6], [7], [10], [11]).…”