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math student
math student
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Let $\Omega_1 \supset \Omega_2 \supset \cdots$ be a sequence of nonempty, open, bounded and convex sets. in $R^n.$

Define $\Omega = \operatorname{int} \Bigl( \overline{\bigcap_{k=1}^{\infty} \Omega_k } \Bigl) $ and suppose that $\Omega$ is nonempty, open, convex and bounded.

I believe that $\partial \Omega_k \rightarrow \partial \Omega$ in the Hausdorff distance. In an article that I am studying, it appears the author is using this fact. I'm not finding this result in the literature, but it probably is a basic result of topology. Can someone indicate to me a book or an article with a proof or just saying that the affirmation is true?

Let $\Omega_1 \supset \Omega_2 \supset \cdots$ be a sequence of nonempty, open, bounded and convex sets.

Define $\Omega = \operatorname{int} \Bigl( \overline{\bigcap_{k=1}^{\infty} \Omega_k } \Bigl) $ and suppose that $\Omega$ is nonempty, open, convex and bounded.

I believe that $\partial \Omega_k \rightarrow \partial \Omega$ in the Hausdorff distance. In an article that I am studying, it appears the author is using this fact. I'm not finding this result in the literature, but it probably is a basic result of topology. Can someone indicate to me a book or an article with a proof or just saying that the affirmation is true?

Let $\Omega_1 \supset \Omega_2 \supset \cdots$ be a sequence of nonempty, open, bounded and convex sets in $R^n.$

Define $\Omega = \operatorname{int} \Bigl( \overline{\bigcap_{k=1}^{\infty} \Omega_k } \Bigl) $ and suppose that $\Omega$ is nonempty, open, convex and bounded.

I believe that $\partial \Omega_k \rightarrow \partial \Omega$ in the Hausdorff distance. In an article that I am studying, it appears the author is using this fact. I'm not finding this result in the literature, but it probably is a basic result of topology. Can someone indicate to me a book or an article with a proof or just saying that the affirmation is true?

replaced deprecated tag 'topology'; rewording
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Ricardo Andrade
Ricardo Andrade
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the validity of a basic affirmationstatement involving the HausdorrfHausdorff distance

Let $\Omega_1 \supset \Omega_2 \supset ...$$\Omega_1 \supset \Omega_2 \supset \cdots$ be a sequence of nonempty  , open, bounded and convex sets.

Define $\Omega = int \Bigl( \overline{\displaystyle\bigcap_{k=1}^{\infty} \Omega_k } \ \Bigl) $$\Omega = \operatorname{int} \Bigl( \overline{\bigcap_{k=1}^{\infty} \Omega_k } \Bigl) $ and supoosesuppose that $\Omega$ is nonempty, open, convex and bounded.

iI believe that $\partial \Omega_k \rightarrow \partial \Omega$ in the Hausdorff distance. In aan article that iI am studying, it appears the author is using this fact. I'm not finding this result in the literature, but it probably is a basic result of topology. Someone can sayCan someone indicate to me a book or aan article with a proof or just saying that the affirmation is true?

the validity of a basic affirmation involving the Hausdorrf distance

Let $\Omega_1 \supset \Omega_2 \supset ...$ a sequence of nonempty  , open, bounded and convex sets.

Define $\Omega = int \Bigl( \overline{\displaystyle\bigcap_{k=1}^{\infty} \Omega_k } \ \Bigl) $ and supoose that $\Omega$ is nonempty, open, convex and bounded.

i believe that $\partial \Omega_k \rightarrow \partial \Omega$ in the Hausdorff distance. In a article that i am studying appears the author is using this fact. I'm not finding this result in the literature, but probably is a basic result of topology. Someone can say to me a book or a article with a proof or just saying that the affirmation is true?

the validity of a basic statement involving the Hausdorff distance

Let $\Omega_1 \supset \Omega_2 \supset \cdots$ be a sequence of nonempty, open, bounded and convex sets.

Define $\Omega = \operatorname{int} \Bigl( \overline{\bigcap_{k=1}^{\infty} \Omega_k } \Bigl) $ and suppose that $\Omega$ is nonempty, open, convex and bounded.

I believe that $\partial \Omega_k \rightarrow \partial \Omega$ in the Hausdorff distance. In an article that I am studying, it appears the author is using this fact. I'm not finding this result in the literature, but it probably is a basic result of topology. Can someone indicate to me a book or an article with a proof or just saying that the affirmation is true?

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math student
math student
  • 269
  • 1
  • 5

the validity of a basic affirmation involving the Hausdorrf distance

Let $\Omega_1 \supset \Omega_2 \supset ...$ a sequence of nonempty , open, bounded and convex sets.

Define $\Omega = int \Bigl( \overline{\displaystyle\bigcap_{k=1}^{\infty} \Omega_k } \ \Bigl) $ and supoose that $\Omega$ is nonempty, open, convex and bounded.

i believe that $\partial \Omega_k \rightarrow \partial \Omega$ in the Hausdorff distance. In a article that i am studying appears the author is using this fact. I'm not finding this result in the literature, but probably is a basic result of topology. Someone can say to me a book or a article with a proof or just saying that the affirmation is true?