Si provano nuovi risultati riguardanti gli «-sets» e gli spazi «Near-compact». Si completano alcune ricerche pubblicate dai primi due autori nel 1978 e si risolvono due problemi recentemente posti da Cammaroto, Gutierrez, Nordo e Prada.
A space is n-arc connected (n-ac) if any family of no more than n-points are contained in an arc. For graphs the following are equivalent: (i) 7-ac, (ii) n-ac for all n, (iii) continuous injective image of a closed subinterval of the real line, and (iv) one of a finite family of graphs. General continua that are ℵ₀-ac are characterized. The complexity of characterizing n-ac graphs for n = 2,3,4,5 is determined to be strictly higher than that of the stated characterization of 7-ac graphs.
Let be the large source of epimorphisms in the category of Urysohn spaces constructed in [2]. A sink is called natural, if for all . In this paper natural sinks are characterized. As a result it is shown that permits no -factorization structure for arbitrary (large) sources.
We investigate notions of -compactness for frames. We find that the analogues of equivalent conditions defining -compact spaces are no longer equivalent in the frame context. Indeed, the closed quotients of frame ‘-cubes’ are exactly 0-dimensional Lindelöf frames, whereas those frames which satisfy a property based on the ultrafilter condition for spatial -compactness form a much larger class, and better embody what ‘-compact frames’ should be. This latter property is expressible without reference...
"Near metric" properties of the space of continuous real-valued functions on a space X with the compact-open topology or with the topology of pointwise convergence are examined. In particular, it is investigated when these spaces are stratifiable or cometrisable.
In 1985, V. G. Pestov described a neighborhood base at the identity of free topological groups on a Tychonoff space in terms of the elements of the fine uniformity on the Tychonoff space. In this paper, we extend Postev’s description to the free paratopological groups where we introduce a neighborhood base at the identity of free paratopological groups on any topological space in terms of the elements of the fine quasiuniformity on the space.