theorem Th59: :: JORDAN2C:75

for n being Nat
for R being Subset of (TOP-REAL n)
for p being Point of (TOP-REAL n)
for P being Subset of (TOP-REAL n) st R is connected & R is open & P = { q where q is Point of (TOP-REAL n) : ( q <> p & q in R & ( for f being Function of I[01],(TOP-REAL n) holds
( not f is continuous or not rng f c= R or not f . 0 = p or not f . 1 = q ) ) ) } holds
P is open