MAT 360 Course Description
In this course we introduce basic concepts of a branch of
mathematics which is usually called "Mathematical Analysis." This is
roughly the study of subsets of Euclidean spaces and properties of
real-valued functions defined on them. Some of these concepts
extend to more general framework of metric spaces.
The main goal of this course is to study the central themes of
"convergence" and "continuity" in an accurate way. To do this,
one has to introduce the key concept of "topology" on the real line
(and in general Rn or metric spaces). This will allow us to define
open and closed sets, the interior, closure and boundary of a set, and
such key concepts as compactness, connectivity, etc. With these
notions in hand, one can begin to study the properties of functions
defined on sets. Many of the standard, intuitive theorems of calculus
will merely become special cases of the theorems proved in this
general setting.
Here is a selection of topics that we will roughly cover in this course:
1. Some history, elements of logic
2. Some elementary set theory
3. The real line, rational, irrational numbers; Euclidean spaces;
metric spaces.
4. Basic Point Set Topology; open, closed, compact,
connected, dense, nowhere dense, and perfect sets. The notion of
completeness
6. Functions defined between metric spaces
7. Continuity; elementary properties of continuous functions
8. Differentiability on the real line; elementary properties of
differentiable functions
9. Sequences and series of functions, uniform convergence
10. Sets of measure zero in analysis
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Math 360