METRIC AND TOPOLOGICAL SPACES I

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Learning Outcomes

define real numbers, identify the convergency and divergency of sequences, explain the limit and continuity of a function at a given point.
construct the geometric model of the set of real numbers.
define the existence of a sequence's limit, if there exists, find the limit.
explain the notion of limit of a function at a given point and if there exists estimate the limit.
define the notion of continuity and obtain the set of points on which a function is continuous.
express the notion of metric space, construct the topology by using the metric notion and using this topology to identify the continuity of the functions which are defined between metric spaces.
explain the notion of metric space.
use the open ball on metric spaces, construct the metric topology and define open-closed sets of the space.
identify the continuity of a function which is defined on metric spaces, at a given point and identify the set of points on which a function is continuous.
define the notion of topology, construct various topologies on a general set which is not empty by using different kinds of techniques, compare these topologies and identify the special subsets of the topology that are called base and subbase which generate elements of the topology.
define the notion of topology.
construct various topologies on a general set, compare them if it is possible.
explain the notion of base and subbase and identify that a subset of a topology is a base or a subbase for this topology.
construct topologies which accept a given family of sets base or subbase.
use the set of functions which are defined on a same set, constructs the weak topology on the domain of these functions.
constructthe subspace topology which is defined on subsets ofthe topological space by using the topology of a given topological space, construct the product topology on the cartesian product of topological spaces by using given two or more topological spaces and construct the quotient topology on a family of sets which is decomposed by an equivalence relation.
define the subspace topology.
construct the product topology on product spaces.
construct the quotient topology.
define and categorize the separation axioms which separate a point from another point, a point from a set that does not contain this point and a set from another set.
express T-1,T-2,T-3 and T-4 separation axioms and use them to prove various properties.
express regularity and normality separation axioms and use them to prove various properties.