Speaker
Description
ntroduction
I In our attempt to answer the question of what Topology is, we would brie
y
journey you through its foundation and formulation. Then you will
appreciate why it stands out as one of the most active areas in all of Maths
today; playing the role as one of the 3 main areas in pure Maths(together
with Algebra and Analysis) and further acting as an indispensable
component of Applied Mathematics.In our world today, it is almost
impossible to understand many real world structures without its use [1].
Remember how you employed the use of quantitative relationships in high
school when you studied ?gures like triangles, angles and circles in
geometry? Well, Topology grew out of geometry by expanding on some of
the ideas of geometry and the loosening of some of the rigid geometrical
structures you disliked. It earned the name rubber-sheet geometry as a
result. So your childhood muse about deforming scaled ?gures may hold in
Topology, Yes! If you do not touch the qualitative properties of that ?gure.
By that, we mean the properties of the ?gure that will remain untouched
after all deformations.
History
I In the eighteenth century, a river called
Pregel
owed through the city of
Konigsberg, dividing it into exactly four
separate regions. Interestingly, seven
bridges crossed the river Pregel and
connected the four regions of the city
as shown in Figure 1. The natural
curiosity of people led to the question
of whether it was possible to take an
entire stroll through the city by
crossing each bridge exactly once?
Leonhard Euler(1707-1783), solved this
long pending problem about the
famous seven bridges in Konigsberg,
Kaliningrad(Russia). Since he was the
rst one to discuss geometry without
measurement, he is considered by many
as the father of Topology. But aside
him are notable mentions like
Descartes, Poincare, Listing, Riemann,
Cantor, Riesz, Frechet etc. To a
topologist, a co?ee cup and a
doughnut are not distinguishable! But
you would be amazed to ?nd out what
a topologist can do [1], [2], [3]. For
more on history see [4].
Credit: [4]
Figure 1: The Konigsberg bridges
Credit: Business Insider
Figure 2: co?ee cup and a doughnut
De?nitions
1).Topology is the mathematical discipline concerned with giving precise
de?nitions for the concept of spatial structure, comparing the various
de?nitions of spatial structure that have been or may be given, and
investigating relations between properties that can be induced into a
topological system [2].
2). Topology is the study of shapes, including their properties, deformations
applied to them, mappings between them and con?gurations composed of
them [1].
3). Topology is the study of all properties of a space that are invariant
under 1:1 bi-continuous mappings [2].
4). Let X be a set. A topology T on X is a collection of subsets of X, each
called an open set, such that;
?; and X are open sets;
?The intersection of ?nitely many open sets is an open set;
?The union of any collection of open sets is an open set.
The set X together with a topology T on X is called a topological space [1].
We shall now proceed to show you one of the most favourite topological
objects known.
The use of (1) and (2), is usually preferred by many to the use of (3) since no prior knowledge base is
required by the reader.
A Favourite Topological Object
Credit:Alon Amit (Quora)
Figure 3: The torus
Note
I The objects we study in Topology are called Topological spaces.
I Simply put, a space is a set with an added structure.
I We may choose to de?ne Topological spaces in terms of open or closed sets.
I A subset A of a topological space X is open if the set X A is closed.
I A subset A of a topological space X is closed if the set X A is open.
I Open sets are considered as \compliments" of closed sets instead of
\opposites".
I A set may be open, closed, half-open or half-closed.
I When the set is half-open or half-closed, it's called clopen.
I Some of the properties of sets that we are usually interested in are its
interior, exterior, closure, boundary and derived sets.
I To deeply appreciate Topology, some key concepts to look out for are
metric spaces, continuity, connectedness and compactness.
Applications of Topology
Topology has many profound applications some of which extend to;
I Embedded systems and Network systems
I Epidemiology
I Data Analysis and Population modelling
I Computer vision
I Manifolds and Cosmology
I Gauge transformations in Physics
I Braids, Knots and DNA
I Medical imaging such as CT Scan
I Fixed Points and Economics
I Geographic information systems
I Motion planning in robotics
I Computer Science
I Phenotype Spaces in Evolutionary Biology etc.
Conclusion
This poster may not be enough to even contain a fraction of the endless
applications of Topology let alone Mathematics. So the next time you
wonder about what Topologists or Mathematicians can do, remind yourself
of their endless viabilities and careers. It is rather unfortunate that we end
our journey here! However, if your interest has been sparked, do continue
to explore further. Who knows, you may become a contributor to this
indispensable form of Mathematics someday.
References
[1] Adams C, Franzosa R. Introduction To Topology, Pure and Applied Pearson Prentice
Hall;2007.
[2] Wolfgang J. Thron Topological Structures Holt, Rinehart and Winston Inc;1966
[3] James Munkres. Topology, Upper Saddle River, N.J . Prentice Hall;2000
[4] O'Connor, JJ and Robertson, EF. Topology in Mathematics, accessed at
http://www-history.mcs.st-andrews.ac.uk/HistTopics/Topology in mathematics
Summary
An overview of Topology is given.
