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My translation of Kuratowski's paper has been removed from docstoc. Some problems arose that location. The paper will reappear elsewhere soon.
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mathematrucker
mathematrucker
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As François mentioned, in 1922 Kuratowski gave the first published example of a space and seed set $x_0$ such that the sequence $\lbrace x_i\rbrace$ is infinite, where $x_{i+1}=x_i\cap cl(cl(x_i)-x_i)$. An English translation of Kuratowski's paper is available here:

http://www.docstoc.com/docs/54437805/On-the-Topological-Closure-Operation

The following equations give a more detailed picture of why the sequence is infinite:

space = $\lbrace1,2,3,\ldots\rbrace$
topology = $\lbrace {\rm space}, \lbrace \rbrace ,\lbrace 1\rbrace ,\lbrace 1,2\rbrace ,\lbrace 1,2,3\rbrace ,\ldots\rbrace $
$x_0=\lbrace 2,4,6,\ldots\rbrace $
$cl(x_0)=\lbrace 2,3,4,\ldots\rbrace $
$y=cl(x_0)-x_0=\lbrace 3,5,7,\ldots\rbrace $
$cl(y)=\lbrace 3,4,5,\ldots\rbrace $
$x_0\cap cl(y)=\lbrace 4,6,8,\ldots\rbrace $
$\ldots$

As François mentioned, in 1922 Kuratowski gave the first published example of a space and seed set $x_0$ such that the sequence $\lbrace x_i\rbrace$ is infinite, where $x_{i+1}=x_i\cap cl(cl(x_i)-x_i)$. An English translation of Kuratowski's paper is available here:

http://www.docstoc.com/docs/54437805/On-the-Topological-Closure-Operation

The following equations give a more detailed picture of why the sequence is infinite:

space = $\lbrace1,2,3,\ldots\rbrace$
topology = $\lbrace {\rm space}, \lbrace \rbrace ,\lbrace 1\rbrace ,\lbrace 1,2\rbrace ,\lbrace 1,2,3\rbrace ,\ldots\rbrace $
$x_0=\lbrace 2,4,6,\ldots\rbrace $
$cl(x_0)=\lbrace 2,3,4,\ldots\rbrace $
$y=cl(x_0)-x_0=\lbrace 3,5,7,\ldots\rbrace $
$cl(y)=\lbrace 3,4,5,\ldots\rbrace $
$x_0\cap cl(y)=\lbrace 4,6,8,\ldots\rbrace $
$\ldots$

As François mentioned, in 1922 Kuratowski gave the first published example of a space and seed set $x_0$ such that the sequence $\lbrace x_i\rbrace$ is infinite, where $x_{i+1}=x_i\cap cl(cl(x_i)-x_i)$.

The following equations give a more detailed picture of why the sequence is infinite:

space = $\lbrace1,2,3,\ldots\rbrace$
topology = $\lbrace {\rm space}, \lbrace \rbrace ,\lbrace 1\rbrace ,\lbrace 1,2\rbrace ,\lbrace 1,2,3\rbrace ,\ldots\rbrace $
$x_0=\lbrace 2,4,6,\ldots\rbrace $
$cl(x_0)=\lbrace 2,3,4,\ldots\rbrace $
$y=cl(x_0)-x_0=\lbrace 3,5,7,\ldots\rbrace $
$cl(y)=\lbrace 3,4,5,\ldots\rbrace $
$x_0\cap cl(y)=\lbrace 4,6,8,\ldots\rbrace $
$\ldots$

Corrected a trivial technical error in my depiction of the topology.
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mathematrucker
mathematrucker
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Indeed the first published solution to the closure-complement-intersection problem appeared in Kuratowski's famous 1922 paper. AsAs François pointed out abovementioned, in 1922 Kuratowski gave anthe first published example of a space and seed set $x_0$ such that the sequence $\lbrace x_i\rbrace$ is infinite, where $x_{i+1}=x_i\cap cl(cl(x_i)-x_i)$. An English translation of Kuratowski's paper is available here:

http://www.docstoc.com/docs/54437805/On-the-Topological-Closure-Operation

The following equations give a more detailed picture of why the sequence is infinite:

space = $\lbrace1,2,3,\ldots\rbrace$
topology = $\lbrace \lbrace \rbrace ,\lbrace 1\rbrace ,\lbrace 1,2\rbrace ,\lbrace 1,2,3\rbrace ,\ldots\rbrace $$\lbrace {\rm space}, \lbrace \rbrace ,\lbrace 1\rbrace ,\lbrace 1,2\rbrace ,\lbrace 1,2,3\rbrace ,\ldots\rbrace $
$x_0=\lbrace 2,4,6,\ldots\rbrace $
$cl(x_0)=\lbrace 2,3,4,\ldots\rbrace $
$y=cl(x_0)-x_0=\lbrace 3,5,7,\ldots\rbrace $
$cl(y)=\lbrace 3,4,5,\ldots\rbrace $
$x_0\cap cl(y)=\lbrace 4,6,8,\ldots\rbrace $
$\ldots$

Indeed the first published solution to the closure-complement-intersection problem appeared in Kuratowski's famous 1922 paper. As François pointed out above, Kuratowski gave an example of a space and seed set $x_0$ such that the sequence $\lbrace x_i\rbrace$ is infinite, where $x_{i+1}=x_i\cap cl(cl(x_i)-x_i)$:

space = $\lbrace1,2,3,\ldots\rbrace$
topology = $\lbrace \lbrace \rbrace ,\lbrace 1\rbrace ,\lbrace 1,2\rbrace ,\lbrace 1,2,3\rbrace ,\ldots\rbrace $
$x_0=\lbrace 2,4,6,\ldots\rbrace $
$cl(x_0)=\lbrace 2,3,4,\ldots\rbrace $
$y=cl(x_0)-x_0=\lbrace 3,5,7,\ldots\rbrace $
$cl(y)=\lbrace 3,4,5,\ldots\rbrace $
$x_0\cap cl(y)=\lbrace 4,6,8,\ldots\rbrace $
$\ldots$

As François mentioned, in 1922 Kuratowski gave the first published example of a space and seed set $x_0$ such that the sequence $\lbrace x_i\rbrace$ is infinite, where $x_{i+1}=x_i\cap cl(cl(x_i)-x_i)$. An English translation of Kuratowski's paper is available here:

http://www.docstoc.com/docs/54437805/On-the-Topological-Closure-Operation

The following equations give a more detailed picture of why the sequence is infinite:

space = $\lbrace1,2,3,\ldots\rbrace$
topology = $\lbrace {\rm space}, \lbrace \rbrace ,\lbrace 1\rbrace ,\lbrace 1,2\rbrace ,\lbrace 1,2,3\rbrace ,\ldots\rbrace $
$x_0=\lbrace 2,4,6,\ldots\rbrace $
$cl(x_0)=\lbrace 2,3,4,\ldots\rbrace $
$y=cl(x_0)-x_0=\lbrace 3,5,7,\ldots\rbrace $
$cl(y)=\lbrace 3,4,5,\ldots\rbrace $
$x_0\cap cl(y)=\lbrace 4,6,8,\ldots\rbrace $
$\ldots$

Removed Facebook link since the page it linked to wasn't being used.
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mathematrucker
mathematrucker
  • 870
  • 6
  • 13

Indeed the first published solution to the closure-complement-intersection problem appeared in Kuratowski's famous 1922 paper. As François pointed out above, Kuratowski gave an example of a space and seed set $x_0$ such that the sequence $\lbrace x_i\rbrace$ is infinite, where $x_{i+1}=x_i\cap cl(cl(x_i)-x_i)$:

space = $\lbrace1,2,3,\ldots\rbrace$
topology = $\lbrace \lbrace \rbrace ,\lbrace 1\rbrace ,\lbrace 1,2\rbrace ,\lbrace 1,2,3\rbrace ,\ldots\rbrace $
$x_0=\lbrace 2,4,6,\ldots\rbrace $
$cl(x_0)=\lbrace 2,3,4,\ldots\rbrace $
$y=cl(x_0)-x_0=\lbrace 3,5,7,\ldots\rbrace $
$cl(y)=\lbrace 3,4,5,\ldots\rbrace $
$x_0\cap cl(y)=\lbrace 4,6,8,\ldots\rbrace $
$\ldots$

Kuratowski's Closure-Complement Problem gives rise to a surprising abundance of interesting problems. Here is a new location for future discussions:

http://www.facebook.com/#!/group.php?gid=114144915263486&ref=ts

Indeed the first published solution to the closure-complement-intersection problem appeared in Kuratowski's famous 1922 paper. As François pointed out above, Kuratowski gave an example of a space and seed set $x_0$ such that the sequence $\lbrace x_i\rbrace$ is infinite, where $x_{i+1}=x_i\cap cl(cl(x_i)-x_i)$:

space = $\lbrace1,2,3,\ldots\rbrace$
topology = $\lbrace \lbrace \rbrace ,\lbrace 1\rbrace ,\lbrace 1,2\rbrace ,\lbrace 1,2,3\rbrace ,\ldots\rbrace $
$x_0=\lbrace 2,4,6,\ldots\rbrace $
$cl(x_0)=\lbrace 2,3,4,\ldots\rbrace $
$y=cl(x_0)-x_0=\lbrace 3,5,7,\ldots\rbrace $
$cl(y)=\lbrace 3,4,5,\ldots\rbrace $
$x_0\cap cl(y)=\lbrace 4,6,8,\ldots\rbrace $
$\ldots$

Kuratowski's Closure-Complement Problem gives rise to a surprising abundance of interesting problems. Here is a new location for future discussions:

http://www.facebook.com/#!/group.php?gid=114144915263486&ref=ts

Indeed the first published solution to the closure-complement-intersection problem appeared in Kuratowski's famous 1922 paper. As François pointed out above, Kuratowski gave an example of a space and seed set $x_0$ such that the sequence $\lbrace x_i\rbrace$ is infinite, where $x_{i+1}=x_i\cap cl(cl(x_i)-x_i)$:

space = $\lbrace1,2,3,\ldots\rbrace$
topology = $\lbrace \lbrace \rbrace ,\lbrace 1\rbrace ,\lbrace 1,2\rbrace ,\lbrace 1,2,3\rbrace ,\ldots\rbrace $
$x_0=\lbrace 2,4,6,\ldots\rbrace $
$cl(x_0)=\lbrace 2,3,4,\ldots\rbrace $
$y=cl(x_0)-x_0=\lbrace 3,5,7,\ldots\rbrace $
$cl(y)=\lbrace 3,4,5,\ldots\rbrace $
$x_0\cap cl(y)=\lbrace 4,6,8,\ldots\rbrace $
$\ldots$

deleted 32 characters in body
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mathematrucker
mathematrucker
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corrected the link at the bottom
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mathematrucker
mathematrucker
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deleted 1 characters in body
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mathematrucker
mathematrucker
  • 870
  • 6
  • 13
Source Link
mathematrucker
mathematrucker
  • 870
  • 6
  • 13