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.

Expert Solution

Colby Messinger answered 2 years ago

Suppose K is a nonempty compact subset of a metric space X and x ∈ X....

Suppose K is a nonempty compact subset of a metric space X and x ∈ X. (i) Give an example of an x ∈ X for which there exists distinct points p, r ∈ K such that, for all q ∈ K, d(p, x) = d(r, x) ≤ d(q, x). (ii) Show, there is a point p ∈ K such that, for all other q ∈ K, d(p, x) ≤ d(q, x). [Suggestion: As a start, let S = {d(x,...

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

Show that if Y is a subspace of X, and A is a subset of Y,...

Show that if Y is a subspace of X, and A is a subset of Y, then the subspace topology on A as a subspace of Y is the same as the subspace topology on A as a subspace of X.

1. Prove that any compact subset ot a metric space is closed and bounded. 2. Prove...

1. Prove that any compact subset ot a metric space is closed and bounded. 2. Prove that a closed subset of a compact set is a compact set.

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.

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...

1. For an m x n matrix A, the Column Space of A is a subspace...

1. For an m x n matrix A, the Column Space of A is a subspace of what vector space? 2. For an m x n matrix A, the Null Space of A is a subspace of what vector space?

Let X be a compact space and let Y be a Hausdorff space. Let f ∶...

Let X be a compact space and let Y be a Hausdorff space. Let f ∶ X → Y be continuous. Show that the image of any closed set in X under f must also be closed in Y .

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.

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