Topological Question: Show that a countable product of a metrizable space is metrizable. (In the product...

Topological Question:

Show that a countable product of a metrizable space is metrizable. (In the product Topology of course)

ie) Show that d is a metric on X that induces the product topology on X

Expert Solution

Colby Messinger answered 2 years ago

Let A be a closed subset of a T4 topological space. Show that A with the...

Let A be a closed subset of a T4 topological space. Show that A with the relative topology is a normal T1

b）Prove that every metric space is a topological space. (c) Is the converse of part (b)...

b）Prove that every metric space is a topological space. (c) Is the converse of part (b) true? That is, is every topological space a metric space? Justify your answer

1)Show that a subset of a countable set is also countable. 2) Let P(n) be the...

1)Show that a subset of a countable set is also countable. 2) Let P(n) be the statement that 13 + 23 +· · ·+n3 =(n(n + 1)/2)2 for the positive integer n. a) What is the statement P(1)? b) Show that P(1) is true, completing the basis step of the proof. c) What is the inductive hypothesis? d) What do you need to prove in the inductive step? e) Complete the inductive step, identifying where you use the inductive hypothesis....

If X is any topological space, a subset A ⊆ X is compact (in the subspace...

If X is any topological space, a subset A ⊆ X is compact (in the subspace topology) if and only if every cover of A by open subsets of X has a finite subcover.

Recall the following definition: For X a topological space, and for A ⊆ X, we define...

Recall the following definition: For X a topological space, and for A ⊆ X, we define the closure of A as cl(A) = ⋂{B ⊆ X : B is closed in X and A ⊆ B}. Let x ∈ X. Prove that x ∈ cl(A) if and only if every neighborhood of x contains a point from A. You may not use any definitions of cl(A) other than the one given.

A topological space X is zero − dimensional if it has a basis B consisting of...

A topological space X is zero − dimensional if it has a basis B consisting of open sets which are simultaneously closed. (a) Prove that the set C = {0, 1}N with the product topology is zero-dimensional. (b) Prove that if (X, d) is a metric space for which |X| < |R|, that is the cardinality of X is less than that of R, then X is zero-dimensional.

1.- let(X1, τ1) and (X2, τ2) are two compact topological spaces. Prove that their topological product...

1.- let(X1, τ1) and (X2, τ2) are two compact topological spaces. Prove that their topological product is also compact. 2.- Let f: X - → Y be a continuous transformation, where X is compact and Y is Hausdorff. Show that if f is bijective then f is a homeomorphism.

The metric space M is separable if it contains a countable dense subset. [Note the confusion...

The metric space M is separable if it contains a countable dense subset. [Note the confusion of language: “Separable” has nothing to do with “separation.”] (a) Prove that R^m is separable. (b) Prove that every compact metric space is separable.

Definition 1 (Topological space). Let X be a set. A collection O of subsets of X...

Definition 1 (Topological space). Let X be a set. A collection O of subsets of X is called a topology on the set X if the following properties are satisfied: (1) emptyset ∈ O and X ∈ O. (2) For all A,B ∈ O, we have A∩B ∈ O (stability under intersection). (3) For all index sets I, and for all collections {Ui}i∈I of elements of O (i.e., Ui ∈ O for all i ∈ I), we have U i∈I...

Prove that the union of infinitely many countable sets is countable.

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