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instructions.html
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<html><head>
<BASE HREF="http://home.cc.umanitoba.ca/~mbell/instructions.html">
<title>Instructions</title></head><body>
<center><h3>
<a href = http://home.cc.umanitoba.ca/~mbell/spacesdef.html>Space Definitions</a>
<br><a href = http://home.cc.umanitoba.ca/~mbell/propsdef.html>Property Definitions</a>
<br><a href = http://home.cc.umanitoba.ca/~mbell/structsdef.html>Structure Definitions</a>
<br><a href = http://www.umanitoba.ca/cgi-bin/math/bell/props.cgi>Return to Property Selection</a>
<br><a href = http://www.umanitoba.ca/cgi-bin/math/bell/structs.cgi>Return to Structure Selection</a>
</h3></center>
<center><h2><b>Spaces, Properties and Structures Instructions</b></h2></center>
A Boolean space is a compact, Hausdorff space which has a basis consisting
of clopen sets.
The spaces of this database are restricted to Boolean non-metrizable spaces which we will
henceforth refer to as "spaces". This is important to <b>emphasize</b>. The program
will simply say that Dyadic implies Ctn 2^W1 whereas what it means is that
every Dyadic non-metrizable space contains the Cantor cube of weight W1.
Another example is that the program will simply say that Orderable and
Chains W are contradictory; what it means is that every orderable Boolean
non-metrizable space has an uncountable chain of clopen sets.
The properties P of this database
are restricted to properties for which there exists a space (see previous) having
P and there exists a space having NOT(P); we will henceforth refer to these
properties as "properties". The structures S of this database
are restricted to structures for which there exists a property (see previous)
having S and there exists a property having NOT(S); we will henceforth refer to these
structures as "structures".
<hr>
<b>Property To Space Selection.</b><br>
On the property list screen there are 2 buttons to the left of
each property. Select the leftmost one to indicate that you want the
property P; select the rightmost one to indicate that you want NOT(P).
After you have made all your selections, select the SUBMIT button. If your
selection contains a pair of properties which are contradictory or
redundant (one follows from the other) and the database knows about it,
then you will be informed of this immediately and must try again.
Otherwise, you will receive a list of solution spaces (in bold) and/or maybe solution
spaces (in italics). A maybe (italic) solution means that, for one of your selected
properties, it is not known to me whether that space has that
property or not, or whether it does or not depends on extra set axioms or
depends on a parameter in the space definition/description that the program cannot
handle. For solutions with a cardinal parameter, it is up to you to determine which
cardinal yields a solution. Spaces with a cardinal parameter have a K in
their name.
Only if there are no ZFC basic solutions and
your property selection is amenable to deducing
extended solutions (i.e., finite sums and/or finite products of spaces in
the database), will such an extended solution (if any) be presented. Lastly
follows all common properties to all solutions (even extended solutions
even when they are not
displayed because there is a ZFC basic solution; this filters out some
obvious properties which could not follow from your selection).
If written in italics,
this means maybe. Common properties that logically follow from exactly one of the
properties that you have selected are not shown. Eg. Separable -> CCC , so
if your selection includes Separable, then CCC will not appear as a common
property. The database knows deductions among pairs of properties (of
course, there are gaps in its knowledge; it currently does not know whether
Homogeneous -> All W-Points).
<p>
If you select a space on the solution screen by clicking to the left of the space,
then the space's properties are
displayed. Custom colors and a background are used to display this
information (so you should turn these features on in your browser's
preferences or else you may get different colors or may not be able to see
some). The colors are as follows:
<dl>
<dt><font color=#ff0000>Red</font>
<dd>I don't know whether the space has the property.
<dt><font color=#0000ff>Blue</font>
<dd>It is consistent with ZFC (or with ZFC+, if the space needs extra set-
theoretic assumptions) that the space has the
property but I don't know if it is consistent with ZFC (or with ZFC+, if
the space needs extra set-theoretic assumptions) that the space doesn't have the property.
<dt><font color=#00ff00>Green</font>
<dd>It is consistent with ZFC (or with ZFC+, if the space needs extra set-
theoretic assumptions) that the space doesn't have the
property but I don't know if it is consistent with ZFC (or with ZFC+, if
the space needs extra set-theoretic assumptions) that the space does have the property.
<dt><font color=#ffff00>Yellow</font>
<dd>It is independent of ZFC (or of ZFC+, if the space needs extra set-
theoretic assumptions) whether the space has or has not the property.
<dt><font color=#ffffff>White</font>
<dd>The space has the property.
<dt><font color=#000000>Black</font>
<dd>The space doesn't have the property.
<dt><font color=#df4798>Purple</font>
<dd>This applies to spaces that have different versions, usually because
the definition depends upon a well-ordering of some set. It means that
one version of the space has the property and one version of the space does
not have the property.
<dt><font color=#b75b00>Brown</font>
<dd>This applies to spaces built from a parameter which is a function of a
cardinal K only. The program distinguishes many shades of brown which would
be too confusing to explicitly display, so it is left to the viewer to
determine for which K's the space has or has not the property. As an
example, one shade is the following: the space has the property if
K is less than or equal to the continuum and the space doesn't have the property
if K is greater than the continuum.
</dl>
<hr>
<b>Structure To Property Selection.</b><br>
This is similar to Property To Space Selection.
<hr>
<b>Lists</b><br>
These are lists of various things.
<hr>
<center><h3>
<a href = http://home.cc.umanitoba.ca/~mbell/spacesdef.html>Space Definitions</a>
<br><a href = http://home.cc.umanitoba.ca/~mbell/propsdef.html>Property Definitions</a>
<br><a href = http://home.cc.umanitoba.ca/~mbell/structsdef.html>Structure Definitions</a>
<br><a href = http://www.umanitoba.ca/cgi-bin/math/bell/props.cgi>Return to Property Selection</a>
<br><a href = http://www.umanitoba.ca/cgi-bin/math/bell/structs.cgi>Return to Structure Selection</a>
</h3></center>
</body>
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