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<html><head>
<BASE HREF="http://home.cc.umanitoba.ca/~mbell/spacesdef.html">
<title>SpacesDefinitions</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/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>Space Definitions</b></h2></center>
<hr>

<dl>
<dt><b>W, W1, W2, W3, C, K</b>
<dd>The first 4 infinite cardinals, the continuum (or the Cantor space or 2^W) and an arbitrary uncountable cardinal.
<dt><b>M, where M is an ordinal or cardinal</b>
<dd>M is given the ordinal topology by default.
<dt><b>DX</b>
<dd>The set X with the discrete topology.
<dt><b>(X)'</b>
<dd>The subspace of X of all non-isolated points.
<dt><b>X x Y</b>
<dd>The product of X and Y.
<dt><b>X^K</b>
<dd>The K power of X.
<dt><b>HX</b>
<dd>The Hyperspace of all non-empty closed subsets of X.
<dt><b>Sup X</b>
<dd>The Superextension of X.
<dt><b>EX</b>
<dd>The Absolute of X.
<dt><b>BX</b>
<dd>The Stone-Cech cptn of X.
<dt><b>X*</b>
<dd>The Stone-Cech remainder BX-X.
<dt><b>Lk X</b>
<dd>The space of all complete subgraphs of the intersection graph of the
    clopen sets of X, with the Tychonov topology.
<dd><cite>M. Bell, C.M.U.C. 23,3 (1982) 525-536.</cite>
<dt><b>Cen X</b>
<dd>The space of all centered subcollections of the clopen sets of X, with
the Tychonov topology.
<dd><cite>M. Bell, Fund. Math. CXXV (1985) 47-58.</cite>
<dt><b>AX</b>
<dd>1 pt cptn of X.
<dt><b>2 Lx O</b>
<dd>2^O with the lexicographic order topology where O is an ordinal.
<dt><b>Path T</b>
<dd>The space of all paths thru the tree T, with the Tychonov topology.
<dd><cite>S. Todorcevic, Studia Math. 116,1 (1995) 49-57.</cite>
<dt><b>Tree T</b>
<dd>The 1 pt cptn of the tree T with the tree topology.
<hr>
<dt><b>2^K</b>
<dt><b>2^K | 2</b>
<dd>Collapse 2 distinct points of 2^K to a point.
<dt><b>2^W3 | W3+1</b>
<dd>Collapse the closed subset of 2^W3 which consists of all functions f
    which are initially 0's followed by all 1's to a point.
<dd><cite>M. Bell, C.M.U.C. 31,4 (1990) 775-779.</cite>
<dt><b>Pas Hom K</b>
<dd>Pasenkov's homogeneous dyadic of weight 2^K which is not a Cantor cube.
<dd><cite>V. Pasenkov, Soviet Math. Dokl. 15,1 (1974) 43-47.</cite>
<dt><b>VDou Rig K</b>
<dd>Van Douwen's rigid dyadic space of weight K.
<dd><cite>E. van Douwen, Top. and its Applns 47 (1992) 203-208.</cite>
<dt><b>Dyad Sum Rig</b>
<dd>A rigid 1 pt cptn of a sum of countably many quite different rigid dyadic spaces.
<hr>
<dt><b>A(K)</b>
<dt><b>Two 1's K</b>
<dd>The subspace of 2^K of all functions with at most 2 1's.
<dt><b>Fin 1's K</b>
<dd>1 pt cptn of the sum of spaces X_n where n &lt; W and X_n is the subspace
    of 2^K of all functions with at most n 1's.
<dd><cite>M. Bell, C.M.U.C. 26,2 (1985) 353-361.</cite>
<dt><b>A(K)^W</b>
<dt><b>A(K)^K</b>
<hr>
<dt><b>(K + 1)^W</b>
<hr>
<dt><b>K + 1</b>
<dt><b>Long C</b>
<dd>Lexicographic product of W1+1 and C.
<dt><b>AA</b>
<dd>The C version of Alexandroff's Double Arrow Line, the W+1 lexicographic power of 2
minus the two isolated points.
<dt><b>2 Lx W*2</b>
<dt><b>2 Lx W^2</b>
<dt><b>2 Lx W1</b>
<dt><b>Aron Line</b>
<dd>An Aronszajn line built from a binary, normal, special, Aronszajn tree.
<dt><b>DeG Rig</b>
<dd>De Groot and Maurice's rigid HL and HS Cor.
<dd><cite>J. de Groot and M. Maurice, P.A.M.S. 19,4 (1968) 844-846.</cite>
<hr>
<dt><b>Path W1</b>
<dd>Use the tree of all functions with domain a countable successor ordinal
    and range W1, ordered by extension.  This tree is NOT a normal tree.
<dt><b>Path Aron</b>
<dd>Use an Aronszajn subtree of the tree of all increasing, well-ordered
    sequences of rationals with a maximal element, ordered by extension.
<dt><b>Path WO(Q)</b>
<dd>Use the tree of all increasing, bounded, well-ordered sequences of
    rationals, ordered by extension.
<dt><b>Path Small K</b>
<dd>Use a tree of small height W and having that distinct nodes have different
cardinalities of successors.  The parameter K is the supremum of these
cardinalities.
<dt><b>Path Big K</b>
<dd>Use a tree of big height W1 and having that distinct nodes have different
cardinalities of successors.  The parameter K is the supremum of these
cardinalities.
<dt><b>Path Tod</b>
<dd>Use Todorcevic's tree T of all closed subsets of W1 contained in a
    bistationary subset S of W1, modified by inserting countable W-ary trees
    between a node and its successors in T, ordered by extension.
<dd><cite>S. Todorcevic, Handbook of Set-Theoretic Topology (1984).</cite>
<dt><b>Tree C</b>
<dd>Use the binary tree of height W+1.
<dt><b>Tree WO(Q)</b>
<dd>Use the tree of all increasing, bounded, well-ordered sequences of
    rationals, ordered by extension.
<dt><b>Tree W1</b>
<dd>Use the binary tree of height W1.
<hr>
<dt><b>C Dub</b>
<dd>Alexandroff duplicate of C.
<dt><b>VDou Dub</b>
<dd>Van Douwen's first countable cptn of the binary tree of height W+1 with
    the tree topology.
<dd><cite>E. van Douwen and J. van Mill, Top. and its Applns 13 (1982) 21-32 </cite>
<dt><b>Bell Dub</b>
<dd>Bell's first countable supercompact space which has VDou Dub
    embedded as a zeroset.
<dd><cite>M. Bell, Colloq. Math. XLIII,2 (1980) 233-241.</cite>
<dt><b>Helly</b>
<dd>Helly's space of non-decreasing functions from C to C.
<dt><b>VMill Hom</b>
<dd>Van Mill's homogeneous Eberlein compact space.
<dd><cite>J. van Mill, Pac. J. of Math. 101,1 (1982) 141-146.</cite>
<dt><b>Pur MN</b>
<dd>Purisch's MN space which is not orderable.
<dd><cite>S. Purisch, ...</cite>
<dt><b>Wat-Wei</b>
<dd>Watson and Weiss's HS and HL space with minimal preimages.
<dd><cite>S. Watson and W. Weiss, Top. and its Applns 28 (1988) 177-179.</cite>
<hr>
<dt><b>Mad W</b>
<dd>The boolean algebra generated in W by the singletons and a MAD family
    of least cardinality a.
<dt><b>Ord W</b>
<dd>The boolean algebra generated in W by the singletons and a tower of
    minimal cardinality t.
<dt><b>Sim W</b>
<dd>Simon's cptn of W whose remainder is the space of all paths thru a tree
    pi-base of W* of minimal height h.
<dd><cite>P. Simon, Soviet Math. Dokl. 19,6 (1978) 1573-1577</cite>
<dt><b>Lad W1</b>
<dd>The 1 pt cptn of a ladder space on Lim(W1) whose rungs are successor
    ordinals.
<dt><b>Sup W</b>
<dt><b>(Sup W)'</b>
<dt><b>Sup W*</b>
<hr>
<dt><b>Lk BW</b>
<dt><b>Lk W*</b>
<dt><b>Lk Sp Graph</b>
<dd>The space of all complete subgraphs of a Sierpinski Graph on an W1-dense subset
    of C of cardinality W1.
<dt><b>Lk Cgt Seq</b>
<dd>The space of all complete subgraphs of the graph of all increasing
    sequences s in C with sup(s) less than 1, where s--t iff sup(s) not
    in t and sup(t) not in s.
<dd><cite>M. Bell, Can. J. Math. XXXV,5 (1983) 824-838.</cite>
<dt><b>Chains W+1</b>
<dd>The space of all chains of the binary tree of height W+1.
<dt><b>Marc W+2</b>
<dd>W. Marciszewski's Strong Eberlein and non-Uniform Eberlein of Cantor-Bendixson height W+2.
<dd><cite>W. Marciszewski, Studia Math. 112 (2) (1995) 189-194.</cite>
<dt><b>Cen BW</b>
<dt><b>Cen W*</b>
<dt><b>Cen 3^W</b>
<dd>Bell's sigma-2-lkd, not sigma-3-lkd, centered space built from a
    rectangle algebra.
<dd><cite>M. Bell, Can. J. Math. XXXV,5 (1983) 824-838.</cite>
<dt><b>Cen Prod W</b>
<dd>Bell's sigma-n-lkd (for all n), non-separable, centered space built
    from a rectangle algebra.
<dd><cite>M. Bell, Topology Proc. 5 (1980) 11-25.</cite>
<dt><b>Cen Tod</b>
<dd>Todorcevic's small, ccc, non-separable, centered space built from a tower of
    minimal cardinality t.  Since a cofinal subset of a tower is again a tower,
    we assume that the tower, when viewed as characteristic functions in C,
    is t-dense in C.
<dd><cite>S. Todorcevic and B. Velickovic, Compositio Math. 63 (1987) 391-408</cite>
<hr>
<dt><b>BW</b>
<dt><b>W*</b>
<dt><b>BK</b>
<dt><b>K*</b>
<dt><b>(W x 2^K)*</b>
<dt><b>Seq UF</b>
<dd>Dow, Gubbi, Szymanski's Stone-Cech cptn of a Sequence space built
    with ultrafilters.
<dd><cite>A. Dow, A. Gubbi, and A. Szymanski, P.A.M.S. 102,3 (1988) 745-748.</cite>
<dt><b>EC</b>
<dt><b>E(2^K)</b>
<dt><b>E(W*)</b>
<dt><b>Leb | Nul</b>
<dd>The boolean algebra of Lebesgue measurable subsets of I modulo the nullsets.
<hr>
<dt><b>H(BW)</b>
<dt><b>(H(BW))'</b>
<dt><b>[H(C)]</b>
<dd>The boolean algebra generated by all closed subsets of 2^W.
<dt><b>[Ctble(C)]</b>
<dd>The boolean algebra generated by all countable subsets of 2^W.
<dt><b>Rect W^W</b>
<dd>The boolean algebra generated by all rectangles in W^W.
<dt><b>Rect 2^W</b>
<dd>The boolean algebra generated by all rectangles in 2^W.
<dt><b>Rect Tod</b>
<dd>Todorcevic's ZFC example of a compact ccc non-separable space with
    countable pi-character. A subalgebra of Rect 2^W.
<dd><cite>S.Todorcevic, Chain-Condition Methods in Topology.</cite>
<dt><b>Rect Evens</b>
<dd>Bell's space of total ideals built from the evens. A subalgebra of Rect 2^W.
<dt><b>Rect Incr W1</b>
<dd>Bell's space of total ideals built from an W1 almost increasing chain. A subalgebra
of Rect 2^W.
<dd><cite>M. Bell, P.A.M.S. 104,2 (1988) 635-640.</cite>
<dt><b>Rect Hau Gap</b>
<dd>Bell's space of total ideals built from the canonical Hausdorff Gap. A subalgebra
of Rect 2^W.
<dd><cite>M. Bell, Lecture Notes in Mathematics 1401, Springer-Verlag (1987) 1-4.</cite>
<hr>
<dt><b>Wat AP</b>
<dd>Watson's almost P-space with no P-points.
<dd><cite>S. Watson, P.A.M.S 123,8 (1995) 2575-2577.</cite>
<dt><b>Scat</b>
<dd>Juhasz and Weiss's thin-tall scattered space (1 pt cptn of).
<dd><cite>I. Juhasz and W. Weiss, Colloq. Math. XL,1 (1978) 63-68.</cite>
<dt><b>Scat Dub</b>
<dd>A first countable version of JW's Scat.
<dd><cite>I. Juhasz and W. Weiss, Colloq. Math. XL,1 (1978) 63-68.</cite>
<hr>
<dt><b>(Fin 1) x C</b>
<dd>(Fin 1's W1) x C
<dt><b>(C+ +1) Lx C</b>
<dt><b>(Tree C) x C</b>
<dt><b>BW x C</b>
<dt><b>AA x AA</b>
<dt><b>(W1+1) x AW</b>
<dt><b>W* x 2^C</b>
<dt><b>(C Dub)^W</b>
<dt><b>(VDou Dub)^W</b>
<dt><b>(W*)^W</b>
<dt><b>W1+1 | 2</b>
<dd>Collapse {W,W1} to a point.
<dt><b>AA(W1)</b>
<dd>Only split W1 points in the Double Arrow AA.
<dt><b>Dub(2^W1|W)</b>
<dd>Double (isolate) a countable dense subset of 2^W1.
<dt><b>A(W1 x C)</b>
<dt><b>A(W1 x 2^W1)</b>
<dt><b>A(DW1 x W*)</b>
<dt><b>(AA(W1))^W</b>
<dt><b>Big Sum Rig</b>
<dd>A rigid 1 pt cptn of a big sum of diverse rigid spaces.
<hr>
<dt><b>A(Meas)</b>
<dd>Meas is the first measurable cardinal.
<dt><b>Kun L</b>
<dd>A Kunen compact L-space.
<dd><cite>K. Kunen, Top. and its Applns 12 (1981) 283-287.</cite>
<dt><b>JKR</b>
<dd>Juhasz, Kunen, and Rudin's S-space (the 1 pt cptn of).
<dd><cite>I. Juhasz, K. Kunen, and M. Rudin, Can. J. Math. 28 (1976) 998-1005</cite>
<dt><b>JKR Dub</b>
<dd>A first countable double of JKR's S-space.
<dd><cite>I. Juhasz, K. Kunen, and M. Rudin, Can. J. Math. 28 (1976) 998-1005</cite>
<dt><b>Sus Line</b>
<dd>A Suslin line built from a binary, normal, Suslin tree.
<dt><b>Path Sus</b>
<dd>The space of all paths thru an W-ary, normal, Suslin tree.
<dt><b>Dow Monolith</b>
<dd>Dow's compact monolithic ccc space with uniform character W1 from Diamond.
<dd><cite>A.Bella and A.Dow, On R-Monolithic Spaces.</cite>
<dt><b>Ost S</b>
<dd>Ostaszewski's S-space (the 1 pt cptn of) from Diamond.
<dd><cite>A. Ostaszewski, J. London Math. Soc. 14,2 (1976) 505-516.</cite>
<dt><b>Fed S</b>
<dd>Fedorchuk's S-space from Diamond.
<dd><cite>V. Fedorcuk, Math. USSR Sbornik 28,1 (1976) 1-26.</cite>
<dt><b>MAC</b>
<dd>Bell's ccc non-separable space from MA(countable posets) and cf(C) = W1.
<dd><cite>M. Bell, Topology Proc. 5 (1980) 11-25.</cite>
<dt><b>Fed No Seq's</b>
<dd>Fedorchuk's small space with no convergent sequences.
<dd><cite>V. Fedorcuk, Math. Proc. of Cambridge Philos. Soc. 81,2 (1977) 177-181.</cite>
<dt><b>Malyhin FU</b>
<dd>Malyhin's FU space with no G_delta points.
<dd><cite>V. Malykin, Math. Notes 41 (1987) 210-216.</cite>
<dt><b>BGT</b>
<dd>The Cohen generic boolean algebra of size W1.
<dd><cite>M. Bell, J. Ginsburg, and S. Todorcevic, Top. and its Applns 14 (1982) 1-12.</cite>
<dt><b>H(BGT)</b>
<dt><b>Coh W2</b>
<dd>The space of all complete subgraphs of the Cohen generic graph on W2.
<dd><cite>M. Bell, C.M.U.C. 23,3 (1982) 525-536.</cite>
<dt><b>Coh W2 2 1's</b>
<dd>The space of at most 2 element subgraphs of the Cohen generic graph on W2.
<dd><cite>M. Bell, C.M.U.C. 23,3 (1982) 525-536.</cite>
<dt><b>Coh Char W</b>
<dd>Bell's First Countable modification of Coh W2 2 1's.
<dd><cite>M. Bell, Top. and its Applns 35 (1990) 153-156.</cite>
<dt><b>Wat Cor</b>
<dd>Watson's Cor with no 2 points of the same character.
<dd><cite>S. Watson, Pac. J. Math. 125,1 (1986) 251-256.</cite>
<hr>
<dt><b>(Ost S) x C</b>
<dt><b>(Kun L)^W</b>
<dt><b>(MAC)^W</b>
<dt><b>Sus GW</b>
<dd>A cptn GW with GW-W homeomorphic to a Sus Line.
</dl>
<center><h3>
<a href = http://home.cc.umanitoba.ca/~mbell/instructions.html>Instructions</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>
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