Linear continuum

Apr 22, 2026 - 19:30
0 0
Linear continuum

link author: James Munkres (via WP:JWB)

← Previous revision Revision as of 13:58, 22 April 2026
Line 11: Line 11:
  • For each ''x'' in ''S'' and each ''y'' in ''S'' with ''x'' < ''y'', there exists ''z'' in ''S'' such that ''x'' < ''z'' < ''y''
    		•
  • For each ''x'' in ''S'' and each ''y'' in ''S'' with ''x'' < ''y'', there exists ''z'' in ''S'' such that ''x'' < ''z'' < ''y''
    		•
    A [[Set (mathematics)|set]] has the least upper bound property, if every nonempty subset of the set that is bounded above has a least upper bound in the set. Linear continua are particularly important in the field of [[topology]] where they can be used to verify whether an [[total order|ordered set]] given the [[order topology]] is [[Connected space|connected]] or not.{{cite book|last=Munkres|first=James|title=Topology, 2nd ed.|year=2000|publisher=[[Pearson Education]]|isbn=0-13-181629-2|pages=31,153}}
    A [[Set (mathematics)|set]] has the least upper bound property, if every nonempty subset of the set that is bounded above has a least upper bound in the set. Linear continua are particularly important in the field of [[topology]] where they can be used to verify whether an [[total order|ordered set]] given the [[order topology]] is [[Connected space|connected]] or not.{{cite book|last=Munkres|first=James|author-link=James Munkres|title=Topology|ed=2nd|year=2000|publisher=[[Pearson Education]]|isbn=0-13-181629-2|pages=31,153}}


    Unlike the standard real line, a linear continuum may be bounded on either side: for example, any (real) [[closed interval]] is a linear continuum.
    Unlike the standard real line, a linear continuum may be bounded on either side: for example, any (real) [[closed interval]] is a linear continuum.
    Line 56: Line 56:


    ==Topological properties==
    ==Topological properties==
    Even though linear continua are important in the study of [[total order|ordered sets]], they do have applications in the mathematical field of [[topology]]. In fact, we will prove that an ordered set in the [[order topology]] is [[Connected space|connected]] if and only if it is a linear continuum. We will prove one implication, and leave the other one as an exercise. (Munkres explains the second part of the proof in {{cite book|last=Munkres|first=James|title=Topology, 2nd ed.|year=2000|publisher=Pearson Education|isbn=0-13-181629-2|pages=153–154}})
    Even though linear continua are important in the study of [[total order|ordered sets]], they do have applications in the mathematical field of [[topology]]. In fact, we will prove that an ordered set in the [[order topology]] is [[Connected space|connected]] if and only if it is a linear continuum. We will prove one implication, and leave the other one as an exercise. (Munkres explains the second part of the proof in {{cite book|last=Munkres|first=James|author-link=James Munkres|title=Topology|ed=2nd|year=2000|publisher=Pearson Education|isbn=0-13-181629-2|pages=153–154}})


    '''Theorem'''
    '''Theorem'''
    Read More