SUO: *Date 01 May 2002 -- Landscape Paintings
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Landscape Paintings
The word "landscape" is notoriously ambiguous --
it can mean a terrain or the pictures thereof --
but I know no place like the Land of Ontology
where the "landscape painters" read their job
description as a writ to scatter their paints
all over the map.
In other words ...
One thing that everybody appears apt to forget or at least gloss over
is the difference between a mathematical model and a physical reality.
Continuity is defined relative to a topology and there is no way that
we can know to an absolute certainty what topology Nature indeed uses.
Most likely the reality is vastly stranger than we would ever imagine!
So the option continuous/discrete, with respect to the model and with
regard to the reality, gives us two approximately independent choices,
not just one. We can use real analysis to do number theory, and just
because one brand of model appears to fit reality best today tells us
little about what sort of model will seem to fit things best tomorrow.
If you know how God topologizes then you already know more than I ever will,
and so I can only bow in silence to the knowledge thereof. Still, my sense
of the depths of my ignorance tells me that the very nature of the relation
between any abstract model in mathematics and any concrete case in reality
is never more than that of one possible approach toward an ideal. And so
I sense that the lines, planes, and spaces of the rational model and the
empirical object are altogteher skew -- that the very nature of this gap
does not permit one to argue from any transient felicity of the model to
a certainty about the ontology. To argue that all features of the model
are necessarily features the object is to mistake a scope for a scape.
Jon Awbrey
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Topology
1. Topological Spaces
1.1. Topologies and Neighborhoods
01. http://suo.ieee.org/ontology/msg03863.html
02. http://suo.ieee.org/ontology/msg03867.html
03. http://suo.ieee.org/ontology/msg03868.html
04. http://suo.ieee.org/ontology/msg03869.html
1.2. Closed Sets
05. http://suo.ieee.org/ontology/msg03870.html
1.3. Accumulation Points
06. http://suo.ieee.org/ontology/msg03871.html
07. http://suo.ieee.org/ontology/msg03872.html
1.4. Closure
08. http://suo.ieee.org/ontology/msg03874.html
09. http://suo.ieee.org/ontology/msg03880.html
1.5. Interior and Boundary
10. http://suo.ieee.org/ontology/msg03882.html
11. http://suo.ieee.org/ontology/msg03883.html
12. http://suo.ieee.org/ontology/msg03888.html
13. http://suo.ieee.org/ontology/msg03889.html
1.6. Bases and Subbases
14. http://suo.ieee.org/ontology/msg03892.html
15. http://suo.ieee.org/ontology/msg03893.html
16. http://suo.ieee.org/ontology/msg03894.html
17. http://suo.ieee.org/ontology/msg03899.html
18. http://suo.ieee.org/ontology/msg03900.html
19. http://suo.ieee.org/ontology/msg03903.html
1.7. Relativization, Separation
20. http://suo.ieee.org/ontology/msg03908.html
21. http://suo.ieee.org/ontology/msg03914.html
22. http://suo.ieee.org/ontology/msg03916.html
23. http://suo.ieee.org/ontology/msg03917.html
1.8. Connected Sets
24. http://suo.ieee.org/ontology/msg03918.html
25. http://suo.ieee.org/ontology/msg03920.html
2. Convergence [skipped for now]
3. Product and Quotient Spaces
26. http://suo.ieee.org/ontology/msg03921.html
3.1. Continuous Functions
27. http://suo.ieee.org/ontology/msg03922.html
28. http://suo.ieee.org/ontology/msg03923.html
29. http://suo.ieee.org/ontology/msg03924.html
30. http://suo.ieee.org/ontology/msg03925.html
All of the above material is excerpted from:
| John L. Kelley, 'General Topology',
| Van Nostrand Reinhold, New York, NY, 1955.
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