Topological Spaces
... When d is the discrete metric and K < 1, this generates the basis of singletons. For R this generates a basis containing c2 = c(< 2c ) elements. We can economise by using the density of Q in R, and restrict the basis to {N (r1 , r2 )} where r1 and r2 are rational (0 < r2 < K), which is a countable b ...
... When d is the discrete metric and K < 1, this generates the basis of singletons. For R this generates a basis containing c2 = c(< 2c ) elements. We can economise by using the density of Q in R, and restrict the basis to {N (r1 , r2 )} where r1 and r2 are rational (0 < r2 < K), which is a countable b ...
1 Weak Topologies
... 3. The collection {fα | α ∈ A} separates points from closed sets in X if whenever B is closed in X and x 6∈ B there exists α ∈ A such that fα (x) 6∈ fα (B). Recall that the continuous functions from a completely regular space X to [0, 1] separate points from closed sets, and Urysohn’s theorem shows ...
... 3. The collection {fα | α ∈ A} separates points from closed sets in X if whenever B is closed in X and x 6∈ B there exists α ∈ A such that fα (x) 6∈ fα (B). Recall that the continuous functions from a completely regular space X to [0, 1] separate points from closed sets, and Urysohn’s theorem shows ...
Sandwich-type characterization of completely regular spaces
... Theorem 1 (Blair [1], Lane [8]). For X a topological space and two arbitrary functions g, h : X → I, the following statements are equivalent : (1) There exists a continuous function f : X → I such that g ≤ f ≤ h. (2) If s < t in I, then [g ≥ t] and [h ≤ s] are completely separated. The equivalence ( ...
... Theorem 1 (Blair [1], Lane [8]). For X a topological space and two arbitrary functions g, h : X → I, the following statements are equivalent : (1) There exists a continuous function f : X → I such that g ≤ f ≤ h. (2) If s < t in I, then [g ≥ t] and [h ≤ s] are completely separated. The equivalence ( ...
Order of Topology
... is a topology on Y, called the subspace topology. With this topology, Y is called a subspace of X; its open sets consist of all intersections of open sets of X with Y. Definition Let X and Y be topological spaces. The product topology on X Y is the topology having as basis the collection B of all ...
... is a topology on Y, called the subspace topology. With this topology, Y is called a subspace of X; its open sets consist of all intersections of open sets of X with Y. Definition Let X and Y be topological spaces. The product topology on X Y is the topology having as basis the collection B of all ...
Topology I Final Solutions
... 2. (a) Give an example of a subspace A of a space X, in which there is a relatively open set S of A that is not an open set of X. (b) Give an example of a closed mapping that is not continuous. (c) Give an example of subsets A and B of R2 such that A and B are disconnected, but A ∪ B is connected. ( ...
... 2. (a) Give an example of a subspace A of a space X, in which there is a relatively open set S of A that is not an open set of X. (b) Give an example of a closed mapping that is not continuous. (c) Give an example of subsets A and B of R2 such that A and B are disconnected, but A ∪ B is connected. ( ...
Part1 - Faculty
... If the neighborhood system U(x) contains the neighborhoods Ui and Uk , then it also contains their intersection Um = Uj ∩ Uk If the neighborhood system U(x) contains the neighborhoods Ui and Uk , then it also contains their union Um = Uj ∪ Uk If Um is a neighborhood of the point x. Then there ...
... If the neighborhood system U(x) contains the neighborhoods Ui and Uk , then it also contains their intersection Um = Uj ∩ Uk If the neighborhood system U(x) contains the neighborhoods Ui and Uk , then it also contains their union Um = Uj ∪ Uk If Um is a neighborhood of the point x. Then there ...
2 Continuous maps. 3 Induced topologies.
... image are open sets in X. This topology is denoted f∗ (U). The formula below defines the direct image topology. V ∈ f∗ (U) ⇔ f −1 (V ) ∈ U ...
... image are open sets in X. This topology is denoted f∗ (U). The formula below defines the direct image topology. V ∈ f∗ (U) ⇔ f −1 (V ) ∈ U ...
