Lecture Notes on Topology for MAT3500/4500 following JR
... Having done this, we can reap some awards. For instance, the definition of what it means for a function f : X → Y , from a topological space X to a topological space Y , to be continuous, is simply: For each open subset V in Y the preimage f −1 (V ) is open in X. This may be compared with the (ǫ, δ ...
... Having done this, we can reap some awards. For instance, the definition of what it means for a function f : X → Y , from a topological space X to a topological space Y , to be continuous, is simply: For each open subset V in Y the preimage f −1 (V ) is open in X. This may be compared with the (ǫ, δ ...
Topological and Limit-space Subcategories of Countably
... incomparable with the maximal topological subcategory identified above). We shall see that one such subcategory arises in a very natural way. Let us consider what happens when equilogical spaces are restricted to equivalence relations over countably based spaces. We say that a topological space is c ...
... incomparable with the maximal topological subcategory identified above). We shall see that one such subcategory arises in a very natural way. Let us consider what happens when equilogical spaces are restricted to equivalence relations over countably based spaces. We say that a topological space is c ...
Chapter 5 Countability and Separation Axioms
... are important examples of Banach spaces. If l∞ = {x = (xn ) : (xn ) is a bounded real sequence } and d∞ (x, y) = sup{|xn − yn | : n ≥ 1}, then (l∞ , d∞ ) is also a metric space. Let X = {x = (xn ) : xn = 0 or xn = 1}. For x, y ∈ X, x 6= y, d(x, y) = 1. Hence (X, d∞ ) (that is d∞ is restricted to th ...
... are important examples of Banach spaces. If l∞ = {x = (xn ) : (xn ) is a bounded real sequence } and d∞ (x, y) = sup{|xn − yn | : n ≥ 1}, then (l∞ , d∞ ) is also a metric space. Let X = {x = (xn ) : xn = 0 or xn = 1}. For x, y ∈ X, x 6= y, d(x, y) = 1. Hence (X, d∞ ) (that is d∞ is restricted to th ...
Domain Theory - School of Computer Science, University of
... A first step towards Domain Theory is the familiar result that every monotone function on a complete lattice, or more generally on a directed-complete partial order with least element, has a least fixpoint. (For an account of the history of this result, see [LNS82].) Some early uses of this result ...
... A first step towards Domain Theory is the familiar result that every monotone function on a complete lattice, or more generally on a directed-complete partial order with least element, has a least fixpoint. (For an account of the history of this result, see [LNS82].) Some early uses of this result ...
DISJOINT UNIONS OF TOPOLOGICAL SPACES AND CHOICE Paul
... Theorem 20.10 p. 147 given in [w] goes through in ZF0 with some minor changes.) However, this conclusion does not hold for metacompact spaces. Dieudonné’s Plank (Example 89, in [ss] p. 108) is an example of a metacompact, non-normal space. Any infinite set X endowed with the discrete topology is an ...
... Theorem 20.10 p. 147 given in [w] goes through in ZF0 with some minor changes.) However, this conclusion does not hold for metacompact spaces. Dieudonné’s Plank (Example 89, in [ss] p. 108) is an example of a metacompact, non-normal space. Any infinite set X endowed with the discrete topology is an ...
Analytic Baire spaces - Department of Mathematics
... Non-separable case). Let X be an analytic space (more generally, paracompact and K-analytic – as de…ned in §2). Then X is a Baire space i¤ X = f (P ) for some continuous, index- -discrete map f on a completely metrizable space P with the property that there exists a dense completely metrizable G sub ...
... Non-separable case). Let X be an analytic space (more generally, paracompact and K-analytic – as de…ned in §2). Then X is a Baire space i¤ X = f (P ) for some continuous, index- -discrete map f on a completely metrizable space P with the property that there exists a dense completely metrizable G sub ...
