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T_α

Last-modified: 2011-11-18 (金) 16:29:36

Definition Edit

  • A topological space (X,τ) is said to be T_α if the following is satisfied;
    1. Let imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20x%5cin%20X%2c%20%5cmbox%7b%20if%20%7d%5coverline%7b%5c%7bx%5c%7d%7d%5cneq%20X%20%5c%5d%7d%25.png then there exists at least one point p such that imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20p%5cin%20X%5csim%5coverline%7b%5c%7bx%5c%7d%7d%20%5cmbox%7b%20and%20%7d%5coverline%7b%5c%7bp%5c%7d%7d=%5c%7bp%5c%7d%20%5c%5d%7d%25.png ;
    2. X contains at least one point x such that imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20%5coverline%7b%5c%7bx%5c%7d%7d=%5c%7bx%5c%7d%20%5c%5d%7d%25.png ;
    3. If C and D are two disjoint closed subsets of X and have disjoint neighbourhoods, then imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20%5coverline%7b%5c%7bx%5c%7d%7d=x%20%5c%5d%7d%25.png for each imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20x%5cin%20C%5ccup%20D%20%5c%5d%7d%25.png.

(For a set S, X〜S is relative complement of S with respect to a set X.)

Property Edit

  • A normal T_α space is a T_4? space.

Reference Edit

  • Y.W. Kim , A note on separation axioms weaker than T1., Kyungpook Math. J. 6 21-26 (1964).