Topology - my thoughts

A large part of mathematics is to see and make generalizations. Topology and the agreed upon definition of it make such a generalization about "open" sets in the Real number space. We have the idea of an "open" set and a "closed" set when describing a range of numbers. We indicate "(3,5)" as open and "[3,5]" as closed. With the open 3,5, we are not including the points 3 and 5 in the range. This means, I can get infinitely close to 3 and infinitely close to 5 but I will never reach there and these numbers are in fact the boundaries. With closed 3,5 we are including the endpoints in the range, meaning, we can get to 3 and 5 but not a step further.

(3,5) = {2.999,2.99,2.98,..., 4.98, 4.99, 4.999} [3,5] = {3, 3.1, 3.2, ..., 4.8, 4.9, 5.0}

This idea of "openness" of a set of points is generalized in topology. A topology is then defined by what kind of structures can we obtain by defining what sets are open and what sets are closed. For example, if we have three points a,b,c, the set containing nothing and the set containing all the points must be open.

Now, to move forward and get an intuitive understanding of how the definitions of a "topology" can be developed, we need to understand how mechanisms are described using set theory. In the language of sets, there are a few interesting things we can do, for example, given two sets we can either take intersections or unions. If we have the sets, (2,4), (3,5), (4,6), their intersection yields the empty set. If we have the sets (2,4) and (3,5) their intersection yields (3,4) which is open. If we take a union for the first example, it yields (2,6) which is the whole range, which is open.

From this little exercise, we have understood that the empty set and the whole set must always be open, and any arbitrary intersection and infinite unions must be open. If we completely drop the initial idea where we drew this from, i.e. open intervals in the real number space, we can go back to our points a,b,c and describe a topological space for them defining what sets are open and what sets are not. However, we are not doing it completely arbirarily, we are bound by the abstraction we have done from the real number space.

Open vs. Closed Sets in ℝ

Consider intervals on the real line:

• Open interval: $(3,5) = {x \in \mathbb{R} \mid 3 < x < 5}$
  Points approach boundaries but never reach 3 or 5.
• Closed interval: $[3,5] = {x \in \mathbb{R} \mid 3 \leq x \leq 5}$
  Endpoints 3 and 5 are included.

Core Abstraction of Topology

The key generalization is defining a topological space by specifying which subsets are considered "open". This must satisfy:

1. $\emptyset$ and the entire space $X$ are open
2. Arbitrary unions of open sets are open: $$\bigcup_{\alpha \in I} U_\alpha \text{ is open}$$
3. Finite intersections of open sets are open: $$\bigcap_{k=1}^n U_k \text{ is open}$$

Motivation from Real Analysis

Consider open intervals in ℝ:

• Intersection: $(2,4) \cap (3,5) = (3,4)$ (open)
• Union: $(2,4) \cup (3,5) \cup (4,6) = (2,6)$ (open)
• Infinite intersection: $\bigcap_{n=1}^\infty \left(3-\frac{1}{n}, 5\right) = [3,5)$ (not open)
  (Shows why only finite intersections are required)

Finite Point Example

For $X = {a, b, c}$, we define a topology by declaring which sets are open, subject to the axioms. For instance:

• Possible topology: $\mathcal{T} = {\emptyset, {a}, {a,b}, {a,b,c}}$
• Valid because:
  • $\emptyset$ and $X$ are open
  • ${a} \cup {a,b} = {a,b}$ (open)
  • ${a} \cap {a,b} = {a}$ (open)

This construction isn't arbitrary—it distills essential properties of open sets in $\mathbb{R}$:

• Boundary behavior (exclusion/inclusion of limits)
• Consistency under set operations
• Preservation of "closeness" without metric dependence

The power of topology lies in how these simple axioms generate profound structures applicable from discrete spaces to infinite-dimensional manifolds.