Boundaries via Distance: Topologically Speaking

A probing mind with queries,
Conditions and consequences,
Nets of thoughts converging
Seeking the solutions fitting.
Mathematical thinking- an idea
Solid and precise, yet so intangible.

Math-a fearsome field to ever so many,
A way of thinking with precision;
A language of mind in focus,
Where logic forms bricks to bolster
To build the structures for giants to stand.

Each mind has a language
To be nurtured, cajoled and pampered,
To lay foundations strong to withstand
The tempests of things to come;
This language is part of mind, so abstract,
Who will dare to keep it there?
Or
Is it beyond a language of mind?

Is this a form of thought endowed
With a philosophy of its own mold?
Does that create a path for mind
To follow what we want to find?
Isn’t one’s thought abstract by nature?
Can anyone see one’s own thought?

So
Being abstract is part of nature.

Why do we fear what we can’t see?
Freedom and beauty of abstractness,
A combo of such rare class!
Encompassing depths of sea.
Epsilon, delta all so clear,
Neighborhoods of countless many,
Bringing all beyond and together,
With different sense of, oh, so near.

Topologists, they say, don’t see a difference,
‘Coffee cups from doughnuts’
Why so, how absurd, are they blind!
Doughnut and coffee cups, how so different!

How close a point is from another
In the structure-the topology- which is created;
Bend it, mold it in a shape so well,
Be careful not to break and leave some.
There is no difference in the substance
Of the new incarnation to the one before!
I have heard of this philosophy
As the essence of nature, the core

As a child is taught to count
A sense of distance forms with it:
How far apart is one from another?
Numbers give this sense as such.
Distance notion builds Calculus,
Analysis and so on,
Structures of limit, continuity, and beyond.
Creating possibilities of countless
Applications, problems’ solutions
Could never have been possible
But for these wings of abstractness!

As we move from number sets
To sets of any, arbitrarily conceived,
Can we have this distance concept?
Yes, we can, here comes the notion,
The space with a metric so defined,
An abstract form of distance,
A set with distance, call it a Metric Space.

The nearness concept in a metric space
Guided by the ‘metric’-
Subtle connection to a numerical notion.
Yet with freedom of structure of sets
And keeping Calculus, Analysis intact,
But rich enough to build on more.

Taking cue from this structure of sets
Topology, created by great minds
Able to see in abstract and beyond.
Topological space- a set with a structure
Defined on the collection of its subsets.
The structure-the Topology-
Tells the nearness of a point from another
Epsilon, delta, so abstract
Pave the way for more fluid
Treatment of continuity, convergence and such
Keeping foundations learned before
Safe and giving room for more.

Beauty of this abstract distance,
Nearness notion so thought of
Without numerical value claiming it,
Free to fly high, yet knows the distance,
Shines so well, all encompassing,
Calculus, Analysis, Metric Spaces and all!
Topology-the abstract distance,
Makes it possible for sets of points
Come together or stay apart.

Boundaries formed, distance comes in,
Positions are taken with choices in,
Philosophies of minds blend in,
A tool with an eye on the goal.
Collectively civilizations fostered,
Wonder the all-embracing nature,
Philosophical key- a notion to nurture?
Topology-a mathematical philosophy raised.

Author’s Note: When work is done in a specific field, within the boundary of a field, the effect of the work does not stay within that boundary. The effect of the research is universal. Topology is one such field within the  universe of mathematics that blurs most boundaries., and therefore, its ripple effects can be noted across all scientific fields.

Supplementary Materials: None.
Author Contributions: N/A
Funding: None.
Acknowledgments: None.
Conflicts of Interest: None.

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