LECTURE NOTES ON DESCRIPTIVE SET THEORY Contents 1
... 2.2. Urysohn space. A Polish metric space is universal if it contains an isometric copy of any other Polish metric space (equivalently, of every countable metric space). Urysohn constructed such a space with a random construction which predates the random graph. Definition 2.16. Suppose that (X, d) ...
... 2.2. Urysohn space. A Polish metric space is universal if it contains an isometric copy of any other Polish metric space (equivalently, of every countable metric space). Urysohn constructed such a space with a random construction which predates the random graph. Definition 2.16. Suppose that (X, d) ...
The sequence selection properties of Cp(X)
... X − An for n ∈ ω. Then it is a clopen γ -cover in X. For each n ∈ ω, let pn : X ω → X be the projection onto the nth clopen γ -cover of X ω . For arbitrary choices pn−1 (Ukn ) ∈ Un coordinate and let Un = {pn−1 (Um )}m∈ω . Then each Un is a−1 (n ∈ ω), take xn ∈ X − Ukn (n ∈ ω), then (xn ) ∈ / n∈ω p ...
... X − An for n ∈ ω. Then it is a clopen γ -cover in X. For each n ∈ ω, let pn : X ω → X be the projection onto the nth clopen γ -cover of X ω . For arbitrary choices pn−1 (Ukn ) ∈ Un coordinate and let Un = {pn−1 (Um )}m∈ω . Then each Un is a−1 (n ∈ ω), take xn ∈ X − Ukn (n ∈ ω), then (xn ) ∈ / n∈ω p ...
On dimension and σ-p.i.c.-functors
... G (A) = G (X) ∩ F (A) for any compact space X and its closed subset A. Proof. The inclusion ⊂ is obvious. As for the inverse inclusion, we shall prove it in several steps. 1. X is finite. Let X = {x1 , . . . , xm , xm+1 , . . . , xn } , A = {x1 , . . . , xm } , xi 6= xj for i 6= j. Take some set Y0 ...
... G (A) = G (X) ∩ F (A) for any compact space X and its closed subset A. Proof. The inclusion ⊂ is obvious. As for the inverse inclusion, we shall prove it in several steps. 1. X is finite. Let X = {x1 , . . . , xm , xm+1 , . . . , xn } , A = {x1 , . . . , xm } , xi 6= xj for i 6= j. Take some set Y0 ...
Tychonoff`s Theorem
... Ea 6= ∅ for any a, then the axiom of choice says there is a function f : P → ∪a∈P Ea with f (a) ∈ Ea . This means that f (a) > a for every a. So, we create a chain a < f (a) < f (f (a)) < · · · . We know this chain has an upper bound since every chain in P has an upper bound. Call it b. Then f (b) ...
... Ea 6= ∅ for any a, then the axiom of choice says there is a function f : P → ∪a∈P Ea with f (a) ∈ Ea . This means that f (a) > a for every a. So, we create a chain a < f (a) < f (f (a)) < · · · . We know this chain has an upper bound since every chain in P has an upper bound. Call it b. Then f (b) ...
An Introduction to Simplicial Sets
... The relations (8), (9), and (10) for X × Y follow from them holding for X and Y . A subsimplicial set of a simplicial set X is a simplicial set Y which satisfies Yn ⊂ Xn for each n ≥ 0, and which inherits the same di ’s and sj ’s. In [3], this is also known as a subcomplex because simplicial sets th ...
... The relations (8), (9), and (10) for X × Y follow from them holding for X and Y . A subsimplicial set of a simplicial set X is a simplicial set Y which satisfies Yn ⊂ Xn for each n ≥ 0, and which inherits the same di ’s and sj ’s. In [3], this is also known as a subcomplex because simplicial sets th ...
On Noether`s Normalization Lemma for projective schemes
... algebraic structures intrisecally connected to their geometric nature. It is the case of, as an example, rings of functions de ned over open subsets of a underlying topological space. In this chapter we are considering richer algebraic structures, such as modules and algebras over a ring. This will ...
... algebraic structures intrisecally connected to their geometric nature. It is the case of, as an example, rings of functions de ned over open subsets of a underlying topological space. In this chapter we are considering richer algebraic structures, such as modules and algebras over a ring. This will ...
Vasile Alecsandri” University of Bac˘au Faculty of Sciences Scientific
... Definition 3.1. A subfamily mX of the power set P(X) of a nonempty set X is called a minimal structure (briefly m-structure) [17], [18] on X if ∅ ∈ mX and X ∈ mX . By (X, mX ), we denote a nonempty set X with a minimal structure mX on X and call it an m-space. Each member of mX is said to be mX -open ...
... Definition 3.1. A subfamily mX of the power set P(X) of a nonempty set X is called a minimal structure (briefly m-structure) [17], [18] on X if ∅ ∈ mX and X ∈ mX . By (X, mX ), we denote a nonempty set X with a minimal structure mX on X and call it an m-space. Each member of mX is said to be mX -open ...
10/3 handout
... Recall that a subset A of (X, τ ) is connected if there are no open sets U , V which disconnect A (i.e. U ∩ V ∩ A = φ, U ∩ A 6= φ, V ∩ A 6= φ, and A ⊆ U ∪ V ). Connectedness is a topological property which is not hereditary, but is preserved under continuous maps, finite products (see Thm 23.6, Munk ...
... Recall that a subset A of (X, τ ) is connected if there are no open sets U , V which disconnect A (i.e. U ∩ V ∩ A = φ, U ∩ A 6= φ, V ∩ A 6= φ, and A ⊆ U ∪ V ). Connectedness is a topological property which is not hereditary, but is preserved under continuous maps, finite products (see Thm 23.6, Munk ...
Open Covers and Symmetric Operators
... refinement of A if for every B ∈ B there exists a A ∈ A such that B ⊂ A. The family B is said to be a precise refinement of A if B and A are indexed by the same index set S and for every s ∈ S Bs ⊂ As . 2.2 Definition. Let A = {As : s ∈ S} be a subfamily of P(X). The star of a set Y ⊂ X with respect ...
... refinement of A if for every B ∈ B there exists a A ∈ A such that B ⊂ A. The family B is said to be a precise refinement of A if B and A are indexed by the same index set S and for every s ∈ S Bs ⊂ As . 2.2 Definition. Let A = {As : s ∈ S} be a subfamily of P(X). The star of a set Y ⊂ X with respect ...