Family of continuous functions defined on complete metric spaces

[alex297]

Homework Statement

Let X and Y be metric spaces such that X is complete. Show that if {f_α(x) : α ∈ A} is a bounded subset of Y for each x ∈ X, then there exists a nonempty open subset U of X such that {f_α(x) : α ∈ A, x ∈ U} is a bounded subset of Y.

Homework Equations

Definition of complete:
A metric space X is called complete if every Cauchy sequence in X converges in X.

Definition of bounded:
A set S is called bounded if ∃ x ∈ A, R>0 such that B(x,R) ⊃ S.

Baire Category Theorem:
Let {U_n} be a sequence of open dense subsets of (X,d), X complete. Then
∪ U_n, 1≤n<∞ is also dense.

The Attempt at a Solution

My friend and I have been working on this problem for a little while now and we're just plain stuck. I've included everything I find relevant above, including the Baire Category Theorem, which I actually don't see how it's relevant but we saw a solution that used it, even though we don't understand the solution. Here it is in case you can make sense of it (I underlined the parts I didn't follow...pretty much all of it):

Fix a point y₀ ∈ Y. For each n≥1, define E_n to be the set of points x ∈ X such that d(f_α(x), y₀)≤n for all α.
Since the f_α's are continuous, E_n is closed. By hypothesis, X is the union of the E_n's. By the Baire Category Theorem, some E_n has nonempty interior, which we take to be U.

Any help would be much appreciated! Thanks!

 

[Dick]

Homework Statement

Let X and Y be metric spaces such that X is complete. Show that if {f_α(x) : α ∈ A} is a bounded subset of Y for each x ∈ X, then there exists a nonempty open subset U of X such that {f_α(x) : α ∈ A, x ∈ U} is a bounded subset of Y.

Homework Equations

Definition of complete:
A metric space X is called complete if every Cauchy sequence in X converges in X.

Definition of bounded:
A set S is called bounded if ∃ x ∈ A, R>0 such that B(x,R) ⊃ S.

Baire Category Theorem:
Let {U_n} be a sequence of open dense subsets of (X,d), X complete. Then
∪ U_n, 1≤n<∞ is also dense.

The Attempt at a Solution

My friend and I have been working on this problem for a little while now and we're just plain stuck. I've included everything I find relevant above, including the Baire Category Theorem, which I actually don't see how it's relevant but we saw a solution that used it, even though we don't understand the solution. Here it is in case you can make sense of it (I underlined the parts I didn't follow...pretty much all of it):

Fix a point y₀ ∈ Y. For each n≥1, define E_n to be the set of points x ∈ X such that d(f_α(x), y₀)≤n for all α.
Since the f_α's are continuous, E_n is closed. By hypothesis, X is the union of the E_n's. By the Baire Category Theorem, some E_n has nonempty interior, which we take to be U.

Any help would be much appreciated! Thanks!

Let's take your problems one at a time. Start with the statement E_n is closed. For a fixed α is the set of points x ∈ X such that d(f_α(x), y₀)≤n closed? What's the relation of E_n to those sets??