Introduction to General Topology
... neighbourhood of x will be any interval of R and the interval itself does not need to be open. Clearly, points in (0, 1) will belong to the closure of H as well as the points in the set {0, 1, 2}. Hence, H = {0} ∪ [1, 2]. 2. Consider (R, Ou ) and B = {n−1 | n ∈ N}. To compute the closure of B it is ...
... neighbourhood of x will be any interval of R and the interval itself does not need to be open. Clearly, points in (0, 1) will belong to the closure of H as well as the points in the set {0, 1, 2}. Hence, H = {0} ∪ [1, 2]. 2. Consider (R, Ou ) and B = {n−1 | n ∈ N}. To compute the closure of B it is ...
Topology Proceedings - topo.auburn.edu
... is either open or closed. Example 1 continued: We fix a one-to-one mapping 9 of Y into I, and define a mapping f of X o:nto I' as follows: f (x) == x for each x E I, and f(x) == g(x) for each x E Y. Then f is a finite-to-one (and hence compact) mapping of the CD-space X onto the space I which is not ...
... is either open or closed. Example 1 continued: We fix a one-to-one mapping 9 of Y into I, and define a mapping f of X o:nto I' as follows: f (x) == x for each x E I, and f(x) == g(x) for each x E Y. Then f is a finite-to-one (and hence compact) mapping of the CD-space X onto the space I which is not ...
Article
... induced fuzzy topological space studied in [7, 8] are based on Theorem 3.2 of [5], which states that “a fuzzy subset A in completely induced fuzzy topological space (X, C(T)) is open iff for each rI, the strong r-cut r(A) is regular open in the topological space (X, T)”. But, incidentally, it has ...
... induced fuzzy topological space studied in [7, 8] are based on Theorem 3.2 of [5], which states that “a fuzzy subset A in completely induced fuzzy topological space (X, C(T)) is open iff for each rI, the strong r-cut r(A) is regular open in the topological space (X, T)”. But, incidentally, it has ...
preopen sets and resolvable spaces
... (i) (X, τ ) contains an open, dense and hereditarily irresolvable subspace. (ii) Every open ultrafilter on X is a base for an ultrafilter on X . (iii) Every nonempty open set is irresolvable. (iv) For each dense subset D of (X, τ ) , int D is dense. (v) For every A ⊆ X , if int A = ∅ then A is nowhe ...
... (i) (X, τ ) contains an open, dense and hereditarily irresolvable subspace. (ii) Every open ultrafilter on X is a base for an ultrafilter on X . (iii) Every nonempty open set is irresolvable. (iv) For each dense subset D of (X, τ ) , int D is dense. (v) For every A ⊆ X , if int A = ∅ then A is nowhe ...
MA651 Topology. Lecture 11. Metric Spaces 2.
... Definition 63.2. Let Y be a metrizable space. A metric d for Y (that is, one that metrics the given topology of Y ) is called complete if every d-Cauchy sequence in Y converges. It must be emphasized that completeness is a property of metrics: One metric for Y may be complete, whereas another metric ...
... Definition 63.2. Let Y be a metrizable space. A metric d for Y (that is, one that metrics the given topology of Y ) is called complete if every d-Cauchy sequence in Y converges. It must be emphasized that completeness is a property of metrics: One metric for Y may be complete, whereas another metric ...
Convex Optimization in Infinite Dimensional Spaces*
... The idea of duality theory for solving optimization problems is to transform the original problem into a "dual" problem which is easier to solve and which has the same value as the original problem. Constructing the dual solution corresponds to solving a "maximum principle" for the problem. This dua ...
... The idea of duality theory for solving optimization problems is to transform the original problem into a "dual" problem which is easier to solve and which has the same value as the original problem. Constructing the dual solution corresponds to solving a "maximum principle" for the problem. This dua ...
DYNAMIC PROCESSES ASSOCIATED WITH NATURAL NUMBERS
... means of the function ψ. If we call x̂ŷ plane the set (ψ(R+ )) , the hyperbolas xy = k (k > 0) of the xy plane with x > 0 and y > 0 are transformed by means of the function ψ × ψ at the x̂ ⊗ ŷ = k̂ “hyperbolas” of the x̂ŷ plane. We will restrict our attention to the points in the x̂ŷ plane that ...
... means of the function ψ. If we call x̂ŷ plane the set (ψ(R+ )) , the hyperbolas xy = k (k > 0) of the xy plane with x > 0 and y > 0 are transformed by means of the function ψ × ψ at the x̂ ⊗ ŷ = k̂ “hyperbolas” of the x̂ŷ plane. We will restrict our attention to the points in the x̂ŷ plane that ...