Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Master
SUBJECT

Code	Subject	
MMA1010	Modern Aspects in Topology and Measure Theory	

Section	Sem.	Hours: C+S+L	Category	Type	
Mathematics	4	2+1+0	speciality	optional	

Teaching Staff in Charge	Assoc.Prof. ANISIU Valeriu, Ph.D., anisiu@math.ubbcluj.ro	
Aims	The course containes more advanced topics in topology and measure theory not included in
the standard course in real analysis

Content	Filters and nets.
Separation axioms (T3,T4).
Product spaces, Tychonov@s theorem.
Compactifications.
Metrizability of topological spaces. 
The Weierstrass-Stone theorem. Classification of some topological spaces using Venn diagrams.  
Convergences for sequences of measurable functions; classical theorems
L^p spaces.  
Fourier series, Dirichlet and Fejer kernels, pointwise convergence.
Real measures, Radon-Nikodym@s theorem.
Derivation for measures, singular measures, applications.
Measure and integration in product spaces, Fubini@s theorem
Topological groups, Haar measures
Hausdorff measures, fractals.

	
Bibliography	1. V. Anisiu: Topologie şi teoria măsurii. Universitatea $Babeş-Bolyai$ Cluj-Napoca, 1995
2. G.B. Folland: Real Analysis. Modern Techniques and their applications. Wiley, 1999
3. H.L. Royden : Real Analysis, 3rd ed, MacMillan, New York, 1988
4. C. George: Exercises in integration. Springer, 1984
5. L.A. Steen, J.A. Seeback: Counterexamples in Topology. Springer, 1978
6. J. Munkres: Topology, 2nd ed. Prentice Hall, 2000
7. C. Swartz:  Measure, integration and function spaces. Word Scientific, 1994
8. P. Kree: Integration et theorie de la mesure. Une approche geometrique. Ellipses, Paris, 1997
9. W. Rudin: Real and Complex Analysis. McGraw-Hill, 1986

	
Assessment	Midterm test and Final Exam.