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While working on a problem related to my research, I had the following query. It pertains to product spaces:

The Question: Let $(X,d_X)$ and $(Y,d_Y)$ be two Polish (Complete separable metric) spaces. Let $A \subseteq X$ and $B \subseteq Y$. Define

$$A^\epsilon := \{x \in X \vert \quad d_X(x,A) < \epsilon\}$$

$$B^\epsilon := \{y \in Y \vert \quad d_Y(y,B) < \epsilon\}$$

Now consider the product space $(X \times Y,d_{XY})$ (warning: possible ambiguity), Define

$$(A \times B)^\epsilon = \{(x,y) \in X \times Y \vert \quad d_{XY}((x,y),A\times B) < \epsilon\}$$

I need to show that $(A \times B)^\epsilon \subseteq A^\epsilon \times B^\epsilon$.

The Doubts and Ambiguities: I gave the warning above because in the problem I am currently working on , I have assumed that $$d_{XY}((x_1,y_1),(x_2,y_2)) = d_X(x_1,x_2) + d_Y(y_1,y_2)$$ Because there are several ways one could define a metric on the product spaces, I need one which generates the product topology(as per G.F Simmons "Topology and Modern Analysis" definition) obviously. So my doubts are:

  1. Is the metric I chose correct in this regard? If so kindly provide a reference or a proof.
  2. Are there other equivalent metrics? If so what are they? (A reference would be appreciated).
  3. My real problem was on how to define a product metric. Could you provide good references to the same other than "Walter Rudin: Real and Complex Analysis", "Patrick Billingsley: Probability and Measure" and "G.F Simmons: Topology and Modern Analysis" (since I have gone through these)?

Once these doubts are cleared, I will be able to solve/disprove the original problem.

Searches: As for searches, I found this which deals with the separability of a product metric and many such searches. But in all of them, they define the product metric and some question pertaining to completeness etc follows. I wish to show that it is independent of how you define it as long as it generates the product topology in my problem.

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Gautam Shenoy
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1 Answers1

2

I managed to solve it. Just for closure, I'll give my solution:

1) The metric I chose was indeed one of several equivalent ones.

2) Yes

3) Not a problem anymore.

Proof: $$(x,y) \in (A\times B)^\epsilon \iff d_{XY}((x,y),(A \times B)) < \epsilon$$

Now $$d_{XY}((x,y),(A \times B)) = \inf_{(\hat{x},\hat{y}) \in A \times B} d_{XY}((x,y),(\hat{x},\hat{y}))$$ $$= \inf_{(\hat{x},\hat{y}) \in A \times B} d_X(x,\hat{x}) + d_Y(y,\hat{y})$$ $$= \inf_{\hat{x} \in A} d_X(x,\hat{x}) + \inf_{\hat{y} \in B} d_Y(y,\hat{y})$$ $$ = d_X(x,A) + d_Y(y,B)$$

Now $$ d_{XY}((x,y),(A \times B)) < \epsilon \iff d_X(x,A) + d_Y(y,B) < \epsilon$$ $$ \Rightarrow d_X(x,A) < \epsilon , \Rightarrow d_Y(y,B) < \epsilon$$ $$\Rightarrow x \in , y \in B^\epsilon$$ $$\iff (x,y) \in A^\epsilon \times B^\epsilon$$

QED

Gautam Shenoy
  • 10,598