View source for Pseudo-arc

A hereditarily-indecomposable [[Snake-like continuum|snake-like continuum]] which contains more than one point.



====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]]].

====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>