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A hereditarily-indecomposable [[Snake-like continuum|snake-like continuum]] which contains more than one point.
 
A hereditarily-indecomposable [[Snake-like continuum|snake-like continuum]] which contains more than one point.
  
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====Comments====
 
====Comments====
One speaks of  "the"  pseudo-arc, since any two are homeomorphic [[#References|[a2]]]. Like the arc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075590/p0755901.png" />, the pseudo-arc is homeomorphic to each of its non-degenerate subcontinua [[#References|[a7]]]. Yet, like the circle, it is homogeneous [[#References|[a1]]]. Its unique feature — necessarily unique — is that  "almost-all continua are pseudo-arcs" ; more precisely, in the [[Hyperspace|hyperspace]] of subcontinua of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075590/p0755902.png" />-cell for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075590/p0755903.png" />, the pseudo-arcs form a residual set [[#References|[a3]]]. All non-degenerate homogeneous snake-like continua are pseudo-arcs [[#References|[a4]]]. Simpler proofs of the fundamental properties, and some generalizations, are developed in [[#References|[a5]]], [[#References|[a6]]], [[#References|[a8]]].
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One speaks of  "the"  pseudo-arc, since any two are homeomorphic [[#References|[a2]]]. Like the arc $[0,1]$, the pseudo-arc is homeomorphic to each of its non-degenerate subcontinua [[#References|[a7]]]. Yet, like the circle, it is homogeneous [[#References|[a1]]]. Its unique feature — necessarily unique — is that  "almost-all continua are pseudo-arcs" ; more precisely, in the [[Hyperspace|hyperspace]] of subcontinua of an $n$-cell for $n\geq2$, the pseudo-arcs form a residual set [[#References|[a3]]]. All non-degenerate homogeneous snake-like continua are pseudo-arcs [[#References|[a4]]]. Simpler proofs of the fundamental properties, and some generalizations, are developed in [[#References|[a5]]], [[#References|[a6]]], [[#References|[a8]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.H. Bing,  "A homogeneous indecomposable plane continuum"  ''Duke Math. J.'' , '''15'''  (1948)  pp. 729–742</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.H. Bing,  "On snake-like continua"  ''Duke Math. J.'' , '''18'''  (1951)  pp. 853–863</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R.H. Bing,  "Concerning hereditarily indecomposable continua"  ''Pacific J. Math.'' , '''1'''  (1951)  pp. 43–51</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R.H. Bing,  "Each homogeneous nondegenerate chainable continuum is a pseudo-arc"  ''Proc. Amer. Math. Soc.'' , '''10'''  (1959)  pp. 345–346</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J. Krasinkiewicz,  "Mapping properties of hereditarily indecomposable continua"  ''Houston J. Math.'' , '''8'''  (1982)  pp. 507–516</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J. Krasinkiewicz,  P. Minc,  "Mappings onto indecomposable continua"  ''Bull. Acad. Polon. Sci.'' , '''25'''  (1977)  pp. 675–680</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  E.E. Moïse,  "An indecomposable plane continuum which is homeomorphic to each of its non-degenerate subcontinua"  ''Trans. Amer. Math. Soc.'' , '''63'''  (1948)  pp. 581–594</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  L. Oversteegen,  E. Tymchatyn,  "On hereditarily indecomposable compacta"  H. Toruńczyk (ed.)  S. Jackowski (ed.)  S. Spiez (ed.) , ''Geometric &amp; Algebraic Topology'' , ''Banach Center Publ.'' , '''18''' , PWN  (1986)  pp. 407–417</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.H. Bing,  "A homogeneous indecomposable plane continuum"  ''Duke Math. J.'' , '''15'''  (1948)  pp. 729–742</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.H. Bing,  "On snake-like continua"  ''Duke Math. J.'' , '''18'''  (1951)  pp. 853–863</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R.H. Bing,  "Concerning hereditarily indecomposable continua"  ''Pacific J. Math.'' , '''1'''  (1951)  pp. 43–51</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R.H. Bing,  "Each homogeneous nondegenerate chainable continuum is a pseudo-arc"  ''Proc. Amer. Math. Soc.'' , '''10'''  (1959)  pp. 345–346</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J. Krasinkiewicz,  "Mapping properties of hereditarily indecomposable continua"  ''Houston J. Math.'' , '''8'''  (1982)  pp. 507–516</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J. Krasinkiewicz,  P. Minc,  "Mappings onto indecomposable continua"  ''Bull. Acad. Polon. Sci.'' , '''25'''  (1977)  pp. 675–680</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  E.E. Moïse,  "An indecomposable plane continuum which is homeomorphic to each of its non-degenerate subcontinua"  ''Trans. Amer. Math. Soc.'' , '''63'''  (1948)  pp. 581–594</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  L. Oversteegen,  E. Tymchatyn,  "On hereditarily indecomposable compacta"  H. Toruńczyk (ed.)  S. Jackowski (ed.)  S. Spiez (ed.) , ''Geometric &amp; Algebraic Topology'' , ''Banach Center Publ.'' , '''18''' , PWN  (1986)  pp. 407–417</TD></TR></table>

Latest revision as of 12:26, 12 April 2014

A hereditarily-indecomposable snake-like continuum which contains more than one point.


Comments

One speaks of "the" pseudo-arc, since any two are homeomorphic [a2]. Like the arc $[0,1]$, the pseudo-arc is homeomorphic to each of its non-degenerate subcontinua [a7]. Yet, like the circle, it is homogeneous [a1]. Its unique feature — necessarily unique — is that "almost-all continua are pseudo-arcs" ; more precisely, in the hyperspace of subcontinua of an $n$-cell for $n\geq2$, the pseudo-arcs form a residual set [a3]. All non-degenerate homogeneous snake-like continua are pseudo-arcs [a4]. Simpler proofs of the fundamental properties, and some generalizations, are developed in [a5], [a6], [a8].

References

[a1] R.H. Bing, "A homogeneous indecomposable plane continuum" Duke Math. J. , 15 (1948) pp. 729–742
[a2] R.H. Bing, "On snake-like continua" Duke Math. J. , 18 (1951) pp. 853–863
[a3] R.H. Bing, "Concerning hereditarily indecomposable continua" Pacific J. Math. , 1 (1951) pp. 43–51
[a4] R.H. Bing, "Each homogeneous nondegenerate chainable continuum is a pseudo-arc" Proc. Amer. Math. Soc. , 10 (1959) pp. 345–346
[a5] J. Krasinkiewicz, "Mapping properties of hereditarily indecomposable continua" Houston J. Math. , 8 (1982) pp. 507–516
[a6] J. Krasinkiewicz, P. Minc, "Mappings onto indecomposable continua" Bull. Acad. Polon. Sci. , 25 (1977) pp. 675–680
[a7] E.E. Moïse, "An indecomposable plane continuum which is homeomorphic to each of its non-degenerate subcontinua" Trans. Amer. Math. Soc. , 63 (1948) pp. 581–594
[a8] L. Oversteegen, E. Tymchatyn, "On hereditarily indecomposable compacta" H. Toruńczyk (ed.) S. Jackowski (ed.) S. Spiez (ed.) , Geometric & Algebraic Topology , Banach Center Publ. , 18 , PWN (1986) pp. 407–417
How to Cite This Entry:
Pseudo-arc. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pseudo-arc&oldid=31625
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article