Connectedness - GMU Math 631 Spring 2011
... Example 17. A space in which components do not equal quasicomponents. Consider S = {0} ∪ {1/n : n ∈ N} with the topology inherited from the real line and X = ({0, 1} × {0}) ∪ (I × {1/n : n ∈ N}) ⊂ I × S. Let p = h0, 0i and q = h1, 0i. Then the component of p is {p} while the quasicomponent of p is { ...
... Example 17. A space in which components do not equal quasicomponents. Consider S = {0} ∪ {1/n : n ∈ N} with the topology inherited from the real line and X = ({0, 1} × {0}) ∪ (I × {1/n : n ∈ N}) ⊂ I × S. Let p = h0, 0i and q = h1, 0i. Then the component of p is {p} while the quasicomponent of p is { ...
IOSR Journal of Mathematics (IOSR-JM)
... x iI Ai. Then, x Ai for some i. Since Ai is -sg* open , there is a sg*-open set Ui such that x Ui and Ui Ai iI Ai. Hence iI Ai is a -sg* -open set. Thus s* is a topology on X. Remark 3.14: If is not regular then the above theorem is not true, that is s* is not a topology ...
... x iI Ai. Then, x Ai for some i. Since Ai is -sg* open , there is a sg*-open set Ui such that x Ui and Ui Ai iI Ai. Hence iI Ai is a -sg* -open set. Thus s* is a topology on X. Remark 3.14: If is not regular then the above theorem is not true, that is s* is not a topology ...
Asymptotic cones - American Institute of Mathematics
... setting. What language should we work with? What can actually be said in the theory of such a metric structure? (We will abbreviate continuous logic by cclogic.) We will work in an arbitrary complete metric space (Y, d) with a distinguished point e, sorts for the balls B(e, n) of center e and radius ...
... setting. What language should we work with? What can actually be said in the theory of such a metric structure? (We will abbreviate continuous logic by cclogic.) We will work in an arbitrary complete metric space (Y, d) with a distinguished point e, sorts for the balls B(e, n) of center e and radius ...
arXiv:math.OA/9901094 v1 22 Jan 1999
... in mind in this note, suppose that E is a graph with no sinks as in [KPR], suppose that X = E ∞ , the infinite path space, and suppose that σ : X → X is the unilateral shift, σ(x1 , x2 , x3 , . . . ) = (x2 , x3 , . . . ). Then σ is a local homeomorphism, and Γ = GE - the groupoid studied in [KPR]. T ...
... in mind in this note, suppose that E is a graph with no sinks as in [KPR], suppose that X = E ∞ , the infinite path space, and suppose that σ : X → X is the unilateral shift, σ(x1 , x2 , x3 , . . . ) = (x2 , x3 , . . . ). Then σ is a local homeomorphism, and Γ = GE - the groupoid studied in [KPR]. T ...
Algebraic Transformation Groups and Algebraic Varieties
... The above lemma also holds when G/H is Stein [13]. Notice that the lemma is a natural generalization of Matsushima’s Theorem in one direction: if G is reductive and G/H is affine, then the lemma implies Ru (H) = 1 and H is reductive. It is natural to explore whether the converse to Lemma 1 holds. For ...
... The above lemma also holds when G/H is Stein [13]. Notice that the lemma is a natural generalization of Matsushima’s Theorem in one direction: if G is reductive and G/H is affine, then the lemma implies Ru (H) = 1 and H is reductive. It is natural to explore whether the converse to Lemma 1 holds. For ...
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... En ’s and a finite number of points in all the remaining En ’s. It is easy to verify that the function f : (Y, τ ) → (Y, G) defined by f (0) = 1 and f (y) = y for all y = 0 is c-upper semicontinuous, since compact sets in (Y, G) are finite, but the graph of f is not closed since (0, 0) ∈ G(f \ G(f ). ...
... En ’s and a finite number of points in all the remaining En ’s. It is easy to verify that the function f : (Y, τ ) → (Y, G) defined by f (0) = 1 and f (y) = y for all y = 0 is c-upper semicontinuous, since compact sets in (Y, G) are finite, but the graph of f is not closed since (0, 0) ∈ G(f \ G(f ). ...
