Set functions.

26 November 2014
16:00
Leobardo Fernández Román
Abstract
A continuum is a non-empty
compact connected metric space.
Given a continuum X let P(X) be the
power set of X. We define the following
set functions:
 
T:P(X) to P(X) given by, for each A in P(X),
T(A) = X \ { x in X : there is a continuum W
such that x is in Int(W) and W does not
intersect A}.
 
K:P(X) to P(X) given by, for each A in P(X)
K(A) = Intersection{ W : W is a subcontinuum
of X and A is in the interior of W}.
 
Also, it is possible to define the arcwise
connected version of these functions.
Given an arcwise connected continuum X:
 
Ta:P(X) to P(X) given by, for each A in P(X),
Ta(A) = X \ { x in X : there is an arcwise
connected continuum W such that x is in
Int(W) and W does not intersect A}.
 
Ka:P(X) to P(X) given by, for each A in P(X),
Ka(A) = Intersection{ W : W is an arcwise
connected subcontinuum of X and A is in
the interior of W}
 
Some properties, examples and relations
between these functions are going to be
presented.
  • Analytic Topology in Mathematics and Computer Science