1550251515-Classical_Complex_Analysis__Gonzalez_

Topology of Plane Sets of Points 123

Examples 1. Let S =IR and let a neighborhood of x 0 E IR be defined by

N(x 0 ) = {x : x ;::: x 0 }. A set AC IR will be called open if each of its points

has a neighborhood contained in A. With this topology IR is a T 0 -space.

2. Let S consists of at least two points, and let D be a trivial topology
   on S. Then (S, D) is a topological space but not a T 0 -space.


3. The complex plane C with the usual topology is a T;-space ( i =
   O, 1, 2, 3, 4), so it is also a regular and normal space.
   Theorem 2.32 The following properties hold:


4. A topological space S is a T 0 -space iff x, y, E S, x '=/:- y, implies that
   {x} '=/:- {y}.


5. A topological space S is a Ti -space iff each singleton { x} C S is a
   closed set.

3. A topological space Sis a T 2 -space iff for any two distinct points x, y E

S there are closed sets Fi and F 2 such that (a) F 1 U F2 = S, (b) x E Fi

but y f. Fi, (c) y E F 2 but x f. F2..

4. If S is a T 2 -space, then S is a T 1 -space, and if S is a Ti -space, then

S is a To-space ..

5. In a Hausdorff space the limit of a sequence is unique.


6. In a Hausdorff space any two disjoint compact subsets can be separated
   by disjoint neighborhoods.


7. Any metric space is a Hausdorff space (with the usual topology).


8. Any Ta-space is a Hausdorff space.


9. Any T 4 -space is a Ta-space.


10. Every compact Hausdorff space is normal.

For corresponding proofs, the reader may consult reference [1] or [5].

Bibliography

1. P. Alexandroff and H. Hopf, Topology, Vol. I, Springer-Verlag, Berlin, 1935.


2. P. Alexandroff and P. Urysohn, Memoire sur les espaces topologiques compacts,
   Verh. Akad. Wetensch. Amsterdam; 14 (1929), 1-96.


3. S. K. Berberian, Introduction to Hilbert Space, Oxford University Press, New
   York, 1961.


4. E.T. Copson, Metric Spaces, Cambridge University Press, Cambridge, 1968.


5. H.F. Cullen, Introduction to General Topology D. C., Heath, Lexington, Mass.,
   1968.


6. Ph. J, Davis, The Schwarz Function and Its Applications, Carus Mathematics
   Monograph 17, Mathematical Association of America, 1974.