Midterm for MATH 5345H: Introduction to Topology October 14, 2013
... to M . Hint: The previous part should be helpful. (d) Let X = {0, 1}. Recall that we showed that the product X ω is uncountable (Theorem 7.7 of Munkres). Let Y consist of the set of subsets of X ω which are countable; that is, Y = {S ⊆ X ω | S is countable} Show that there exists a bijection from Y ...
... to M . Hint: The previous part should be helpful. (d) Let X = {0, 1}. Recall that we showed that the product X ω is uncountable (Theorem 7.7 of Munkres). Let Y consist of the set of subsets of X ω which are countable; that is, Y = {S ⊆ X ω | S is countable} Show that there exists a bijection from Y ...
PDF
... topology τ generated by all the open singletons makes X a door space: Proof. If B ⊆ X does not contain x, it is the union of elements in A, and therefore open. If x ∈ B, then its complement B c does not, so is open, and therefore B is closed. Since τ = P (A) ∪ {X}, the space X not discrete. In addit ...
... topology τ generated by all the open singletons makes X a door space: Proof. If B ⊆ X does not contain x, it is the union of elements in A, and therefore open. If x ∈ B, then its complement B c does not, so is open, and therefore B is closed. Since τ = P (A) ∪ {X}, the space X not discrete. In addit ...
Theorem: let (X,T) and (Y,V) be two topological spaces... E={G×H:GT,HV} is a base for some topology X×Y.
... Theorem: let (X,T) and (Y,V) be two topological spaces then the collection E={G×H:GT,HV} is a base for some topology X×Y. Definition: let (X,T) and (Y,V) be two topological spaces then the topology W whose base is E is called the product topology for X×Y and (X×Y , W) is called the product of X an ...
... Theorem: let (X,T) and (Y,V) be two topological spaces then the collection E={G×H:GT,HV} is a base for some topology X×Y. Definition: let (X,T) and (Y,V) be two topological spaces then the topology W whose base is E is called the product topology for X×Y and (X×Y , W) is called the product of X an ...
PDF
... A topological vector space is a pair (V, T ), where V is a vector space over a topological field K, and T is a topology on V such that under T the scalar multiplication (λ, v) 7→ λv is a continuous function K × V → V and the vector addition (v, w) 7→ v + w is a continuous function V × V → V , where ...
... A topological vector space is a pair (V, T ), where V is a vector space over a topological field K, and T is a topology on V such that under T the scalar multiplication (λ, v) 7→ λv is a continuous function K × V → V and the vector addition (v, w) 7→ v + w is a continuous function V × V → V , where ...
PDF
... elements of B. We also have the following easy characterization: (for a proof, see the attachment) Proposition. A collection of subsets B of X is a basis for some topology on X if and only if each x ∈ X is in some element B ∈ B and whenever B1 , B2 ∈ B and x ∈ B1 ∩ B2 then there is B3 ∈ B such that ...
... elements of B. We also have the following easy characterization: (for a proof, see the attachment) Proposition. A collection of subsets B of X is a basis for some topology on X if and only if each x ∈ X is in some element B ∈ B and whenever B1 , B2 ∈ B and x ∈ B1 ∩ B2 then there is B3 ∈ B such that ...
What Is...a Topos?, Volume 51, Number 9
... on the Ui ’s which coincide on the intersections Ui ∩ Uj . Now, let C be a category having finite projective limits. To give a topology (sometimes called a Grothendieck topology) on C means to specify, for each object U of C , families of maps (Ui → U)i∈I , called covering families, enjoying propert ...
... on the Ui ’s which coincide on the intersections Ui ∩ Uj . Now, let C be a category having finite projective limits. To give a topology (sometimes called a Grothendieck topology) on C means to specify, for each object U of C , families of maps (Ui → U)i∈I , called covering families, enjoying propert ...