The topology of a closable topological subspace S is closely related to two key sets derived from its closure S‾:

1. The Boundary of S: S‾∖S 2. The Exterior of S: X∖S‾

Consider S=(0,1) in X=R

• Closure: S‾=[0,1] • Boundary: S‾∖S={0,1} • Exterior: X∖S‾=(−∞,0)∪(1,∞) • Topology of S: o S is open because it does not include its boundary points. o Sequences in S can approach 0 and 1, affecting convergence behavior. o The exterior points are entirely separate, influencing global properties

like connectedness.

The topology of a closable subspace S is fundamentally linked to its boundary and exterior: • Boundary Points: Influence whether S is open or closed and affect

local topological properties. • Exterior Points: Define how S sits within the larger space X and affect

global topological properties. Understanding these relationships helps characterize the behavior of S within X, including aspects like continuity, convergence, openness, closedness, and separation from other subsets.

Dear Dr. Sureshkumar Somayajula , thank you very much for your meaningful answer. I hope this answer can iniciate some interesting research - and not only mine or yours.

Thank you Dr. Alexey Orlovsky. I am not Dr yet. Started doing research on Mathematics and Computer science related.

Dear Sureshkumar Somayajula , I think the main question is how mutually heritable (or mutually inducible) these topologies are. I still can't decide on this question. And what do you think about it?

To know my opinion on this question, see my Research proposal Research Proposal How is the topology of a closable topological subspace S rel...

Dear Alexey Orlovsky, Thank you for your thoughtful question. When considering the "mutual heritability" or "mutual inducibility" of topologies, we're essentially looking at whether two topological structures can inherit properties from one another or influence each other. This mutual relationship would mean that the presence or properties of one topology could affect the characteristics or behavior of the other. In our case, analyzing these relationships could shed light on how different topological features interact within a dataset or system, especially in complex, multi-dimensional data like we see in TDA. I'm still exploring this concept myself, as the implications could vary depending on the nature of the dataset and the specific topological features we’re examining. What are your thoughts on potential methods to measure or evaluate this kind of mutual relationship?

Dear Sureshkumar Somayajula, for today, I can say only the following. I believe that mathematics is the physics of semantic space. And the semantic space is nothing but a union of the expansions of the physical space in which we live. Mathematical objects in such a space quite physically affect each other. So far so. And then - we will think.