We show that the extent of a Tychonoff space of countable weak extent can be arbitrary big. The extent of X is e(X) = sup{|F | : F ⊂ X is closed and discrete} while we(X) = min{τ : for every open cover U of X there is A ⊂ X such that |A| ≤ τ and St(A,U) = X} is the weak extent of X (also called the star-Lindelöf number of X). Also we show that the extent of a normal space with countable weak ex...

Assuming Jensen’s principle ♦, there is a compact Hausdorff space X which is hereditarily Lindelöf, hereditarily separable, and connected, such that no closed subspace of X is both perfect and totally disconnected. The Proper Forcing Axiom implies that there is no such space. The ♦ example also fails to satisfy the CSWP (the complex version of the Stone-Weierstrass Theorem). This space cannot c...

The note contains two examples of function spaces Cp(X) endowed with the pointwise topology. The first example is Cp(M),M being a planar continuum, such that Cp(M) is uniformly homeomorphic to Cp(M) if and only ifm = n. This strengthens earlier results concerning linear homeomorphisms. The second example is a non-Lindelöf function space Cp(X), whereX is a monolithic perfectly normal compact spa...

ζ(0 + it) = O(|t| 1 2 + ), for each > 0. Apply the Phragmén–Lindelöf Theorem to ζ(s) in a small interval containing 1. Clearly, we can make the linear function arising from the Phragmén–Lindelöf Theorem to pass arbitrarily close to 0 at σ = 1. Thus, as |t| → ∞, ζ(1 + it) = O(|t| ), for each > 0. To apply the Phragmén–Lindelöf Theorem to ζ(s) in the strip 0 ≤ σ ≤ 1, we set k(σ) = aσ + b. Thus, k...

threats are one of the most fundamental problems of our modern societies that daily will affect citizens in urban spaces. this becomes more problematic when the people in their everyday living spaces encounter with behaviors that cause lack of security and should have been feel comfort and their families used these spaces freely. this causes many problems for citizens. in today cities because o...

A space X is said to be set star-Lindelöf if for each nonempty subset of and collection U open sets in such that ⊆⋃U, there a countable V ⊆ St (⋃V,U). The class spaces lie between the Lindel öf spaces. In this paper, we investigate relationship other related by providing some suitable examples study topological properties

We show that a set of reals is undetermined in Galvin’s point-open game iff it is uncountable and has property C′′, which answers a question of Gruenhage. Let X be a topological space. The point-open game G(X) of Galvin [G] is played as follows. Black chooses a point x0 ∈ X, then White chooses an open set U0 3 x0, then B chooses a point x1 ∈ X, then W chooses an open set U1 3 x1, etc. B wins th...

Every lower semi-continuous closed-and-convex valued mapping Φ : X → 2 , where X is a Σ-product of metrizable spaces and Y is a Hilbert space, has a single-valued continuous selection. This improves an earlier result of the author.

We give an example of a compact space X whose iterated continuous function spaces Cp(X), CpCp(X), . . . are Lindelöf, but X is not a Corson compactum. This solves a problem of Gul’ko (Problem 1052 in [11]). We also provide a theorem concerning the Lindelöf property in the function spaces Cp(X) on compact scattered spaces with the ω1th derived set empty, improving some earlier results of Pol [12...