All roads lead to Rome. But I must say, this journey into topology was embarked from a pavement. Without wasting much time on my kitschy yet romantic exposition of how we got there, let’s dive in into Topology. I am mostly following the Introduction to Topology by Kinsey. Why? ’Cause my advisor told so, that’s why. I am listening to this slowed down bollywood with added rain effects (idk it’s soothing) while I do so: link.

The textbook starts with a quintessential example of turning a doughnut into a coffee cup. Wish I could also pass on all those extra calories. Topology is essentially the study of special properties of surface that are invariant to squishing, squeezing, bending, and stretching. But don’t break it, god forbid!

Two objects are topologically identical if there is a continuous deformation from one to another. In that way, it could be considered ’’rubber-sheet geometry” apparently. This topological equivalence is also referred to as being homeomorphic.

That being said, we start with the first chapter, the simpler of all, the Point-set topology.

Chapter 1: Point-set topology

The study of the most general possible object: a set of points. For now, we will constrain to the real n-space, $\mathbb R^n$.

A disc is typically used to discuss a neighborhood, but this is a misnomer. It essentially represents a $n-$dimensional sphere that is the neighborhood defined in terms of the Euclidean distance.

The interior, exterior, and boundary or limit points are determined depending on the nature of their neighborhood.

The closure $Cl(A)$ is the set of all limit points of $A$ and is a closed set.

A function $f: X\to Y$ is a homeomorphism iff it is continuous, invertible, and its inverse function $f^{-1}$ is also continuous. The spaces $X$ and $Y$ are topologically equivalent.

Topological equivalence is an equivalence relation, meaning it’s symmetric, reflexive, and transitive.

A set of points is compact if every infinite sequence of points in $A$ has a limit point in $A$.

• The real line is not compact since the sequence ${1, 2, 3, 4, \cdots}$ consists of points in $\mathbb R$ but has no limit point in $\mathbb R$. Furthermore, $(0, 1)$ is also not compact.

Intuitively and by Heine-Borel theorem, a compact set is both closed and bounded.

Accordingly, a cube is compact.

Have gone through multiple chapters in the book. Mostly these chapters were on point-set topology and connectedness, product spaces, and quotient spaces. These are not my main focus but I acknowledge that some kind of foundation on them is useful. With that said, I switch into the chapter on Surfaces.

Chapter 2: Surfaces

A topological complex could be constructed by glueing the similarly labeled edges together while being cognizant of the edge directions. For example, for glueing, both the edges should be in the same direction.

A cylinder can be constructed by glueing one set of opposite edges together.

A torus can be constructed by gluing both the opposite edges of a rectangle.

A disk with a zipper, i.e., both the semicircles having the same direction, is topologically equivalent to a sphere’s surface.

There are ofcourse, many different planar diagrams for any surface.

On the other hand, to represent a mobius strip, the opposite edges to be glued in the rectangle should be pointing in the opposite directions so that one of the edges have to be twisted so as to be glued to the other. The mobius strip however, has only one side, in contrast to the cylinder.

A manifold is a topological space such that every point has a neighborhood topologically equivalent to a $n-$ dimensional open disk with center $\mathbf{x}$ and radius $r$. A two-dimensional manifold is often called a surface.

This implies manifolds are Hausdorff spaces.

In classifying the surfaces, the ability to “enclose a cavity” will turn out to be a distinguishing feature.

Removing two discs from two tori and gluing them together to obtain a 2-handled torus. Notice that the planar polygon for an orientable surface is a $4g$ sided polygon, given that $g$ is the genus.

While it might seem odd that for a homeomorphism, cutting is not allowed, any cut can be repaired by gluing things back just the way they were.

Classification of surfaces

Every compact connected surface is homeomorphic to a sphere, a connected sum of $n$ tori, or a connected sum of $n$ projective planes. The steps involved in classifying the surface are as follows.

Step 1: Build a planar model of the surface

Since $S$ is a compact surface, there is a simplical complex on it with finitely many triangles. Since $S$ is connected, its triangles can be rearranged so that each triangle is glued to an earlier one. Then, assemble them in the chosen order to form a polygon representing a planar diagram of the surface.

A surface is connected iff a triangulation can be re-arranged in the order $T_1, T_2, …, T_n$ such that each $T_i$ has atleast one edge common with $T_{i-1}$.

Step 2: A shortcut

A string of edges that occur twice in exactly the same order, taking into account the directions of the edges, we can relabel to consider the string as a single edge.

Note that the edges can occur in two forms: opposing edges or twisted pairs.

Step 3: Eliminate adjacent opposing pairs

Adjacent opposing pairs can be eliminated by foldiing them in and giving them edges together.

Step 4: Eliminate all but one vertext

Step 5: Collecting twisted pairs

A twisted pair of edges labeled $a$ may be made adjacent by cutting along the dotted line and regluing along the original edge $a$.

Step 6: Collecting pairs of opposing pairs

If steps 1 through 5 have been performed, then any opposing pairs must occur in pairs.