clarifiation about topological spaces and more

Question

I am learning topology by myself, and this is a list of my open questions

(when I say $st.R^n$ I mean standard topology on $R^n$)

1. beyond the definition, what actually is a topological space? is it any a set of points which is arranged in a certain way geometrically(like a line, shape, plane etc., not neccessarily euclidian)? if yes, then is $(\{(x,y)|x^2+y^2=r^2\}, st.R)$ for example the topological space of a circle? does every shape in euclidian geometry or manifold have $st.R^n$?

2. can abstaract topological spaces with a topology different than $st.R^n$ be visualised? do they appear anywhere in the real world or are they purely theoretic?

3. when topology is introduced we usually talk about continous deformations, are these actually just homeomorphisms?

4. I know open sets are subsets of the topology on the set, and that is defines the way that points are "connected", but how does it do that?

5. how would you describe things like hexagonal tiling a a topological space(i.e. an ordered pair $(X,)$)?

Answer 1

1. A topological space is a set $X$ and a subset $\mathcal{O}_X\subseteq\mathcal{P}(X)$ of its power set. (I am not sure if the english expression "set system" is used, in german a subset of the power set is called "Mengensystem".) An element $U\in\mathcal{O}_X$, which is a subset $U\subset X$ is called an open set of $X$.

2. If you consider the real world to be the normed vector spaces $\mathbb{R}^3$ or $\mathbb{R}^4$, we always get the same induced topology as all norms on $\mathbb{R}^n$ are equivalent and therefore always the same sets are open.

3. For topological spaces $(X,\mathcal{O}_X)$ and $(Y,\mathcal{O}_Y)$, a map $f\colon X\rightarrow Y$ between their respective sets is called continuous, if preimages of open sets are open: \begin{equation} \forall U\in\mathcal{O}_X\colon f^{-1}(U)\in\mathcal{O}_Y. \end{equation} A homeomorphism between topological spaces is a continuous bijective map with continuous inverse. (Compare this with the definition of a diffeomorphism for example.) Homeomorphic spaces basically have the same respective set and same topology on it, when you just rename everything (which the homeomorphism does).

4. For example, you can ask yourself if for every two point $x,y\in X$, do you have open sets $U,V\in\mathcal{O}_X$ with $x\in U$, $y\in V$ and $U\cap V=\emptyset$? A space like this is called a Hausdorff space, which is easy to remember as the two points can be housed off by disjoint open sets. You can look at similar conitions (weaker and stronger), which are called the seperation axioms. Hausdorff spaces (seperation axiom $T_2$) are the most important among them.

5. Hexagonal tiling is not used for topological spaces as far as I know, but triangulation. You basically take a collection of subsets of a topological spaces, that are homeomorphic to a triangle in $\mathbb{R}^2$, so the verticies and edges overlap.

Answer 2

1. A topological space is nothing more or nothing less than its definition, which is based upon open sets. This being said, topological spaces bear the same relation with classical $\mathbb{R}^2$ and $\mathbb{R}^3$ geometry, as, let's say, groups with arithmetics: that's the cradle of the concept. But like any math concept, it is born to leave its cradle and explore remote places. Like groups are omnipresent in maths, topological spaces are everywhere. For example there is a demonstration of the infinitude of prime numbers, that uses a specific topology on $\mathbb{N}$.

Saying that topology is about "sets of points that are arranged in a certain way geometrically (like a line, shape, plane)" is somewhat misleading: notions of line and plane require a vector space and usually imply a metric; topology is more general than that. Topology allows to extend important notions such as function continuity, to spaces that, e.g., do not have any metric. Such as the vector space of functions from $\mathbb{R}$ to $\mathbb{R}$ with pointwise convergence topology: it cannot be given a metric that preserves its topology.

2. As an example, Zarisky topologies are defined by their closed sets, which are algebraic varieties (solutions of a system of polynomial equations). We may still be in $\mathbb{R}^n$ or $\mathbb{C}^n$, so it can still be visualized, but the closed sets are different from those of standard topology on those spaces.

3. Strictly speaking, "continuous deformations" refer to homotopies, not homeomorphisms. Homotopy is defined between functions; then a derived concept, homotopy equivalence (or homotopy types) is defined between sets.

Homeomorphism is actually a stronger property than homotopy equivalence; i.e. you can have two spaces that are homotopy equivalent, but not homeomorphic, but homeomorphism implies homotopy equivalence.

4. Topology deals with properties of space that do not require a distance. It does that by specifying "open sets". While their definition sounds abstract, you can actually easily express notions such as separation with them. There are actually many different concepts of separation. The most commonly used is: two points are separated if you can find two open sets, each open set contains one of these points, and their intersection is empty.

A space is connected if it cannot be partitioned into two disjoint non-empty open sets. This is tricky because openness is not an absolute, but a relative notion, and here the subsets are considered open relatively to their superset. E.g. $S = [0, 1] \cup [2, 3]$ is disconnected because it is the union of $[0, 1]$ and $[2, 3]$, which are both open subsets of S. (And they are also closed subsets of S and of $\mathbb{R}$, of course).

Like for separation, there are various non-equivalent concepts for connectedness.

5. Assuming you mean the tiling with edges and vertices, not with full-surface hexagons. Then you could define open sets by all open-ended connected paths between any two points of the tiling, and their unions. That would be equivalent to the restriction of open sets of the plane, to the tiling.

Another possibility, more specific to the tiling, would be to define closed sets as edges, and finite sets of edges. Open sets being the complementary sets. That would mean two points in the same edge (that are not vertices) would not be separated. Which is adequate if we think of the tiling primarily as a graph.