Hyperspace of finite unions of convergent sequences

The symbol S(X) denotes the hyperspace of finite unions of convergent sequences in a Hausdor˛ space X. This hyper-space is endowed with the Vietoris topology. First of all, we give a characterization of convergent sequence in S(X). Then we consider some cardinal invariants on S(X), and compare the character, the pseudocharacter, the sn-character, the so-character, the network weight and cs-network weight of S(X) with the corresponding cardinal function of X. Moreover, we consider rank k-diagonal on S(X), and give a space X with a rank 2-diagonal such that S(X) does not Gδ -diagonal. Further, we study the relations of some generalized metric properties of X and its hyperspace S(X). Finally, we pose some questions about the hyperspace S(X).

Cardinal Functions of Purely Atomic Measures

Let $$\mu $$ μ be a purely atomic finite measure. Without loss of generality we may assume that $$\mu $$ μ is defined on $${\mathbb {N}}$$ N , and the atoms with smaller indexes have larger masses, that is $$\mu (\{k\})\ge \mu (\{k+1\})$$ μ ( { k } ) ≥ μ ( { k + 1 } ) for $$k\in {\mathbb {N}}$$ k ∈ N . By $$f_\mu :[0,\infty )\rightarrow \{0,1,2,\dots ,\omega ,{\mathfrak {c}}\}$$ f μ : [ 0 , ∞ ) → { 0 , 1 , 2 , ⋯ , ω , c } we denote its cardinal function $$f_{\mu }(t)=\vert \{A\subset {\mathbb {N}}:\mu (A)=t\}\vert $$ f μ ( t ) = | { A ⊂ N : μ ( A ) = t } | . We study the problem for which sets $$R\subset \{0,1,2,\dots ,\omega ,{\mathfrak {c}}\}$$ R ⊂ { 0 , 1 , 2 , ⋯ , ω , c } there is a measure $$\mu $$ μ such that $$R=\text {rng}(f_\mu )$$ R = rng ( f μ ) . We are also interested in the set-theoretic and topological properties of the set of $$\mu $$ μ -values which are obtained uniquely.

Cardinal functions of the hyperspace of convergent sequences

The symbol 𝓢c(X) denotes the hyperspace of all nontrivial convergent sequences in a Hausdorff space X. This hyperspace is endowed with the Vietoris topology. In the current paper, we compare the cellularity, the tightness, the extent, the dispersion character, the net weight, the i-weight, the π-weight, the π-character, the pseudocharacter and the Lindelöf number of 𝓢c(X) with the corresponding cardinal function of X. We also answer a question posed by the authors in a previous paper.

A new cardinal function on topological spaces

<p>Using neighbourhood assignments, we introduce and study a new cardinal function, namely GCI(X), for every topological space X. We shall mainly investigate the spaces X with finite GCI(X). Some properties of this cardinal in connection with special types of mappings are also proved.</p>

Models of countable theories

We define and study types of a complete first-order theory T. This concept allows us to refine our analysis of Mod(T). If T has few types, then Mod(T) contains a uniquely defined smallest model that can be elementarily embedded into any structure of Mod(T). We investigate the various properties of these small models in Section 6.3. In Section 6.4, we consider the “big” models of Mod(T). For any theory, the number of types is related to the number of models of the theory. For any cardinal κ, I(T, κ) denotes the number of models in Mod(T) of size κ. We prove two basic facts regarding this cardinal function. In Section 6.5, we show that if T has many types, then I(T, κ) takes on its maximal possible value of 2κ for each infinite κ. In Section 6.6, we prove Vaught’s theorem stating that I(T, ℵ0) cannot equal 2. All formulas are first-order formulas. All theories are sets of first-order sentences. For any structure M, we conveniently refer to an n-tuple of elements from the underlying set of M as an “n-tuple of M.” The notion of a type extends the notion of a theory to include formulas and not just sentences. Whereas theories describe structures, types describe elements within a structure. Definition 6.1 Let M be a ν-structure and let ā = (a1, . . . , an) be an n-tuple of M. The type of ā in M, denoted tpM(ā), is the set of all ν-formulas φ having free variables among x1, . . . , xn that hold in M when each xi in is replaced by ai. More concisely, but less precisely: If ā is an n-tuple, then each formula in tpM(ā) contains at most n free variables but may contain fewer. In particular, the type of an n-tuple contains sentences. For any structure M and tuple ā of M, tpM(ā) contains Th(M) as a subset. The set tpM(ā) provides the complete first-order description of the tuple ā and how it sits in M.