Countable Dense Homogeneous Filters
... Take a non-meager P -filter F and two countable dense subsets D0 , D1 . We construct a sequence of partial homeomorphisms hk : P(n(k)) → P(n(k)), where {n(k) : k < ω} is an increasing sequence. Each hk+1 will “extend” hk and the homeomorphism h : P(ω) → P(ω) is the limit of the hk . Intuitively, in ...
... Take a non-meager P -filter F and two countable dense subsets D0 , D1 . We construct a sequence of partial homeomorphisms hk : P(n(k)) → P(n(k)), where {n(k) : k < ω} is an increasing sequence. Each hk+1 will “extend” hk and the homeomorphism h : P(ω) → P(ω) is the limit of the hk . Intuitively, in ...
A Topological Study of Tilings
... Two commonly-used topologies on cellular automata Proximity between tilings and cellular automata Adapt these topologies to tilings Understand the behavior of the space of tilings Study computation from a geometrical point of view ...
... Two commonly-used topologies on cellular automata Proximity between tilings and cellular automata Adapt these topologies to tilings Understand the behavior of the space of tilings Study computation from a geometrical point of view ...
Convergence Classes and Spaces of Partial Functions
... (xn ) is a net in X, then a property will be said to hold eventually with respect to (xn ) if it holds for all n ≥ m for some element m of the index set of (xn ). One point on which we will be specific, however, is in our use of the term “subnet”, and we will adopt Kelley’s definition throughout (se ...
... (xn ) is a net in X, then a property will be said to hold eventually with respect to (xn ) if it holds for all n ≥ m for some element m of the index set of (xn ). One point on which we will be specific, however, is in our use of the term “subnet”, and we will adopt Kelley’s definition throughout (se ...
COUNTABLE DENSE HOMOGENEITY AND THE DOUBLE ARROW
... definition of the middle-thirds Cantor set, it is easy to see that every neighborhood of a point of X contains some interval from {Jn : n < ω}. Thus, by continuity 1−i there exist m < ω and clopen intervals I0 and I1 such that I0 ⊂ Jki , I1 ⊂ Jm and h[I0 ] ⊂ I1 . By Proposition 3.1, we may assume tha ...
... definition of the middle-thirds Cantor set, it is easy to see that every neighborhood of a point of X contains some interval from {Jn : n < ω}. Thus, by continuity 1−i there exist m < ω and clopen intervals I0 and I1 such that I0 ⊂ Jki , I1 ⊂ Jm and h[I0 ] ⊂ I1 . By Proposition 3.1, we may assume tha ...
Compact groups and products of the unit interval
... Remark 2-5. Theorem 2-4 is the best possible in the following sense: there exists a group G with w(G) = No, but S(G) = 1. The compact group T has this property, as T contains 0, but T does not contain a copy of I2. This is so since dim (T) = 1 and dim (I2) = 2. Further, observe that for each positiv ...
... Remark 2-5. Theorem 2-4 is the best possible in the following sense: there exists a group G with w(G) = No, but S(G) = 1. The compact group T has this property, as T contains 0, but T does not contain a copy of I2. This is so since dim (T) = 1 and dim (I2) = 2. Further, observe that for each positiv ...
PROOF. Let a = ∫X f dµ/µ(X). By convexity the graph of g lies
... proper clopen subset.) (X , T ) is said to be path-connected if for any two points x 0 , x 1 ∈ X there exists a continuous curve c : [0, 1] → X with c(i ) = x i . A connected component is a maximal connected subset of X . (X , T ) is said to be totally disconnected if every point is a connected comp ...
... proper clopen subset.) (X , T ) is said to be path-connected if for any two points x 0 , x 1 ∈ X there exists a continuous curve c : [0, 1] → X with c(i ) = x i . A connected component is a maximal connected subset of X . (X , T ) is said to be totally disconnected if every point is a connected comp ...
COMPACT! - Buffalo
... because the sum of the lengths of the deleted intervals is ∑ n = 1; i.e., its measure is n =1 3 zero. On the other hand it has the same size as the entire interval [0,1]. This is strengthened by the problem which appears in W. Rudin's textbook, and on some Ph.D. Qualifying Exams: 2.2. EXERCISE. Each ...
... because the sum of the lengths of the deleted intervals is ∑ n = 1; i.e., its measure is n =1 3 zero. On the other hand it has the same size as the entire interval [0,1]. This is strengthened by the problem which appears in W. Rudin's textbook, and on some Ph.D. Qualifying Exams: 2.2. EXERCISE. Each ...
Cantor`s Theorem and Locally Compact Spaces
... a nonempty open set V1 not containing x1 by the first part, letting U = X. We construct a sequence V1 ⊃ V2 . . . of non-empty closed subsets, by the rule: if Vn is constructed, take a nbhd Vn+1 contained in Vn such that Vn+1 does not include xn+1 . Then the inclusion of closures follows from A ⊂ B → ...
... a nonempty open set V1 not containing x1 by the first part, letting U = X. We construct a sequence V1 ⊃ V2 . . . of non-empty closed subsets, by the rule: if Vn is constructed, take a nbhd Vn+1 contained in Vn such that Vn+1 does not include xn+1 . Then the inclusion of closures follows from A ⊂ B → ...
COUNTABLE PRODUCTS 1. The Cantor Set Let us constract a very
... If we continue in this way, at each stage deleting the open middle third of each closed interval remaining from the previous stage we obtain a descending sequence of closed sets G1 ⊃ G2 ⊃ G3 ⊃ · · · ⊃ Gn ⊃ . . . T The Cantor Set G is defined by G = ∞ n=1 Gn , and being the intersection of closed set ...
... If we continue in this way, at each stage deleting the open middle third of each closed interval remaining from the previous stage we obtain a descending sequence of closed sets G1 ⊃ G2 ⊃ G3 ⊃ · · · ⊃ Gn ⊃ . . . T The Cantor Set G is defined by G = ∞ n=1 Gn , and being the intersection of closed set ...
Cantor set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883.Through consideration of this set, Cantor and others helped lay the foundations of modern point-set topology. Although Cantor himself defined the set in a general, abstract way, the most common modern construction is the Cantor ternary set, built by removing the middle thirds of a line segment. Cantor himself only mentioned the ternary construction in passing, as an example of a more general idea, that of a perfect set that is nowhere dense.