
Examine when the four-set collection {∅, A, B, X} is a topology on X, showing it works if A and B partition X or if one contains the other.
Examine constructing topologies on the set X = {A, B, C} with exactly four members, using partitions of X by A and B and cases of total order by inclusion.
Explore discrete and indiscrete topologies on finite sets, defining discrete topology as the power set and comparing coarser and finer topologies by subset inclusion; illustrated with {a,b,c} and {1,2,3}.
Explain how the intersection of two topologies remains a topology, while the union of two topologies need not be a topology, with examples and key properties.
Explore the subspace topology: define subspace from a subset Y of a topological space X, with open sets as intersections of Y with X's open sets; illustrated with examples.
Demonstrates that if y is a subspace of x with y ⊆ z ⊆ x, the subspace topology on y from x equals that from z.
Define open sets in a topology, compare subspaces, and prove that unions of open sets are open and finite intersections of open sets are open, with examples from discrete topology.
Explore closed sets in topology by defining them as complements of open sets and study examples, including discrete topology. Examine arbitrary intersections and finite unions of closed sets.
Define the closure of a subset as the intersection of all closed supersets, yielding the smallest closed superset containing the subset.
How to become a pro in the topology of mathematics?
This is a high-level mathematics course in the topology of mathematics that is being taught to intermediate-level students in different universities around the world. The course requires basic knowledge of mathematics before enrolling. it is actually, a totally different and unique course in mathematics. We will construct the different topological models of mathematics in this course. By definition, the Topology of Mathematics is actually the twisting analysis of mathematics. Moreover, the topology of mathematics is a high-level math course that is the sub-branch of functional analysis.
We shall discuss the twisting analysis of different mathematical concepts. The course is highly perfect for those who want to explore new concepts in mathematics. We shall start from the basics to advance in this course. Most students which are math lovers demand this course many times and that's why I have constructed this course.
There is a total of 7 sections in this course of 7.5 hours of videos contents. All the videos have been captured on a writing tablet and contain high-definition digital content. We shall discuss every concept with examples and prove the different theorems step by step in a detailed way.
I shall just suggest that, if you have a notebook and pencil during watching the videos, they must note the important concepts during each lecture.
The contents of this course are
Definition of Topology
Examples of Topology
Discrete and In Discrete Topology
Coarser and Finer Topology
Definition and Examples of Subspace
Theorems in Topology
Open Sets
Closed Sets
Closure of a Set in Topology
Dense Set in Topology
Neighborhood Concept in Topology
Limit Point
Interior and Exterior Point
And much more