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propsdef.html
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<html><head>
<BASE HREF="http://home.cc.umanitoba.ca/~mbell/propsdef.html">
<title>PropertiesDefinitions</title></head><body>
<center><h3>
<a href = http://home.cc.umanitoba.ca/~mbell/instructions.html>Instructions</a>
<br><a href = http://home.cc.umanitoba.ca/~mbell/spacesdef.html>Space 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>Property Definitions</h2></center>
<hr>
<dl>
<dt><b>Dyadic</b>
<dd>An image of a Cantor cube.
<dt><b>Polyadic</b>
<dd>An image of an Alexandroff cube.
<dt><b>Scadic</b>
<dd>An image of a product of compact scattered spaces.
<dt><b>Centered</b>
<dd>An image of a space Cen(S) (the space of all centered subcollections of S). The
spaces Cen(S)'s are also known as Adequate compact spaces.
<dt><b>Hyadic</b>
<dd>An image of a hyperspace H(X) of a compact space X.
<dt><b>W* Image</b>
<dd>An image of the space W* of all free ultrafilters on W.
<hr>
<dt><b>Onto BW</b>
<dd>Maps onto the space BW of all ultrafilters on W.
<dt><b>Onto H(BW)</b>
<dd>Maps onto H(BW).
<dt><b>Onto 2^W1</b>
<dd>Maps onto 2^W1.
<dt><b>Onto W1+1</b>
<dd>Maps onto ordinal space W1+1.
<hr>
<dt><b>Contain 2^W1</b>
<dd>Contains the Cantor cube of weight W1.
<dt><b>Contain W1+1</b>
<dd>Contains the ordinal space W1+1.
<dt><b>Contain BW</b>
<dd>Contains BW.
<dt><b>Contain AW</b>
<dd>Contains a non-trivial convergent sequence AW.
<hr>
<dt><b>Strong Eber</b>
<dd>A Strong Eberlein compact space.
<dt><b>Uniform Eber</b>
<dd>A Uniform Eberlein compact space.
<dt><b>Eberlein</b>
<dd>An Eberlein compact space.
<dt><b>Corson</b>
<dd>A Corson compact space.
<dt><b>Cor</b>
<dd>Compact ORderable.
<hr>
<dt><b>Homogeneous</b>
<dd>Homogeneous.
<dt><b>Rigid</b>
<dd>A crowded space whose only autohomeomorphism is the identity.
<dt><b>Cmpn 2</b>
<dd>Supercompact, i.e., has a binary closed subbase.
<dt><b>Cmpn Inf</b>
<dd>Infinite cpctness number. Does not have an n-ary closed subbase, for any n.
<hr>
<dt><b>Monolithic</b>
<dd>Monolithic.
<dt><b>D-separable</b>
<dd>Has a dense W Discrete subspace.
<hr>
<dt><b>Density W</b>
<dd>Separable.
<dt><b>W Linked</b>
<dd>The open sets are the union of countably many linked subcollections.
<dt><b>Property K</b>
<dd>Every uncountable open collection has an uncountable linked subcolln.
<dt><b>CCC</b>
<dd>All disjoint clopen families are countable.
<dt><b>Chains W</b>
<dd>Every chain of clopen sets is countable.
<hr>
<dt><b>Pi Weight W</b>
<dd>Countable pi-weight.
<dt><b>W Dsjt Pi-Bs</b>
<dd>Has a pi-base which is the union of countably many disjoint subcollections.
<dt><b>Pt W Pi-Bs</b>
<dd>Has a point-countable pi-base.
<dt><b>Tree Pi-Base</b>
<dd>Has a pi-base which is a tree under reverse inclusion satisfying that incomparable elements are disjoint.
<hr>
<dt><b>Card Small</b>
<dd>Has cardinality at most the continuum C.
<dt><b>Weight W1</b>
<dd>Has weight W1.
<dt><b>Weight Small</b>
<dd>Has weight at most C.
<hr>
<dt><b>Char W</b>
<dd>First countable.
<dt><b>Frechet-Urys</b>
<dd>Frechet-Urysohn.
<dt><b>Radial</b>
<dd>Replace convergent sequence in Frechet-Urys by a convergent well-ordered net.
<dt><b>Tightness W</b>
<dd>Countable tightness.
<dt><b>Seq'lly Cpct</b>
<dd>Every sequence has a convergent subsequence.
<dt><b>Pi Char W</b>
<dd>Countable pi-character.
<hr>
<dt><b>MNormal</b>
<dd>Monotonically normal.
<dt><b>HLindelof</b>
<dd>Hereditarily Lindelof.
<dt><b>HParacompact</b>
<dd>Hereditarily paracompact.
<dt><b>HNormal</b>
<dd>Hereditarily normal.
<dt><b>HRealcompact</b>
<dd>Hereditarily realcompact.
<dt><b>HSeparable</b>
<dd>Hereditarily separable.
<dt><b>Pt Normal</b>
<dd>Complements of every point are normal.
<hr>
<dt><b>ED</b>
<dd>Extremally disconnected.
<dt><b>BD</b>
<dd>Basically disconnected.
<dt><b>F-Space</b>
<dd>Disjoint open F_sigma's have disjoint closures.
<dt><b>Almost P</b>
<dd>A crowded space with non-empty zerosets having non-empty interior.
<hr>
<dt><b>W Discrete</b>
<dd>Is the union of countably many discrete subspaces.
<dt><b>Scattered</b>
<dd>Every subspace has an isolated point in the subspace topology.
<hr>
<dt><b>Unif Pi Char</b>
<dd>All points have the same pi character.
<dt><b>Crowded</b>
<dd>No isolated points in the space.
<dt><b>All W-Points</b>
<dd>Every non-isolated point is the limit of a non-trivial convergent sequence.
<dt><b>G-Delta Pt</b>
<dd>Has a point with a countable local base.
<dt><b>Weak P Pt</b>
<dd>Has a non-isolated point that is not in the closure of any disjoint countable set.
</dl>
<center><h3>
<a href = http://home.cc.umanitoba.ca/~mbell/instructions.html>Instructions</a>
<br><a href = http://home.cc.umanitoba.ca/~mbell/spacesdef.html>Space 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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