Vipul: New page: {{variation of|topological space}} ==Definition== A '''proximity space''' is a set $X$ along with a binary relation $\delta$ on the power set of $X$ (cal...

2008-11-24T20:02:59Z

New page: {{variation of|topological space}} ==Definition== A '''proximity space''' is a set <math>X</math> along with a binary relation <math>\delta</math> on the power set of <math>X</math> (cal...

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{{variation of|topological space}}

==Definition==

A '''proximity space''' is a set <math>X</math> along with a binary relation <math>\delta</math> on the power set of <math>X</math> (called a ''proximity relation'' or ''nearness relation'') satisfying the following conditions (note that we say that <math>A</math> and <math>B</math> are ''near'', or <math>A \delta B</math>, if they are related, and we say that <math>A</math> and <math>B</math> are ''separated'', or <math>A \not \delta B</math> if <math>A</math> and <math>B</math> are not related):

# '''Intersecting subsets are near''': If <math>A \cap B \ne \varnothing</math>, then <math>A \delta B</math>. In other words, any two intersecting subsets are near.
# '''Near implies nonempty''': The empty set is not near to any set. In other words, <math>A \delta B</math> implies that both <math>A</math> and <math>B</math> are nonempty.
# '''Symmetry''': <math>A \delta B \iff B \delta A</math>.
# '''Distributivity''': <math>A \delta (B \cup C)</math> if and only if <math>A \delta B</math> or <math>A \delta C</math>.
# '''Separation''': If <math>A \not \delta B</math>, there exists a set <math>C</math> such that <math>A \not \delta C</math> and <math>B \not \delta (X \setminus C)</math>.

Proximity space - Revision history

Vipul: New page: {{variation of|topological space}} ==Definition== A '''proximity space''' is a set X along with a binary relation \delta on the power set of X (cal...

Vipul: New page: {{variation of|topological space}} ==Definition== A '''proximity space''' is a set $X$ along with a binary relation $\delta$ on the power set of $X$ (cal...