Cantor set

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Cantor set is a <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> of points lying on a single <a href="/facts/Line_segment/V8E0M4qk">line segment</a> that has a number of unintuitive properties. It was discovered in 1874 by <a href="/facts/Henry_John_Stephen_Smith/sD12Jf6L">Henry John Stephen Smith</a> and mentioned by German mathematician <a href="/facts/Georg_Cantor/k5v1DvW4">Georg Cantor</a> in 1883.
Through consideration of this set, Cantor and others helped lay the foundations of modern <a href="/facts/Point-set_topology/ezC7B0yP">point-set topology</a>. The most common construction is the Cantor ternary set, built by removing the middle third of a line segment and then repeating the process with the remaining shorter segments. Cantor mentioned this ternary construction only in passing, as an example of a <a href="/facts/Perfect_set/wKUjfCkM">perfect set</a> that is <a href="/facts/Nowhere_dense_set/m0oF4X8R">nowhere dense</a>.
More generally, in topology, a <a href="/facts/Cantor_space/h1sNqjBT">Cantor space</a> is a topological space <a href="/facts/Homeomorphism/jMfWg3Mn">homeomorphic</a> to the Cantor ternary set (equipped with its subspace topology). The Cantor set is naturally <a href="/facts/Homeomorphism/jMfWg3Mn">homeomorphic</a> to the countable product 
 
 
 
 
 
 
 2
 _
 
 
 
 
 N
 
 
 
 
 
 {\displaystyle {\underline {2}}^{\mathbb {N} }}
 
 of the <a href="/facts/Discrete_two-point_space/kdTMqJoW">discrete two-point space</a> 
 
 
 
 
 
 2
 _
 
 
 
 
 {\displaystyle {\underline {2}}}
 
. By a theorem of <a href="/facts/L._E._J._Brouwer/6q6ULf3b">L. E. J. Brouwer</a>, this is equivalent to being perfect, nonempty, compact, <a href="/facts/Metrizable_space/BCLdo5Jf">metrizable</a> and zero-dimensional.

Cantor set open-in-new

Cantor set