How is ''a topology'' related to the topology that study fundamental property of shape?

Question

$T$ is “a topology” of $X$ if the following 4 conditions are satisfied.

1. The empty set is in $T$.

2. $X$ is in $T$.

3. The intersection of a finite number of sets in $T$ is also in $T$.

4. The union of an arbitrary number of sets in $T$ is also in $T$.

From above, it seems to me that a topology is an attempt to generalize and talk about convergence and closeness without involving actual metric structure. However, I have no clue as to how the topology defined above is useful in studying the fundamental property of shape. Loop number and knots etc.

I want a brief explanation about how “a topology” defined above is related to the toplogy that talks about shape.

Answer 1

Topology that "talks about shape" is the study of continuous and bicontinuous (continuous forward and backwards) functions. The notion of equivalence under continuous deformation is captured in homeomorphisms (bicontinuous bijections).

The definition of continuous function that is usually introduced in point-set topology is that the preimage of an open set is open. For homeomorphism, we map not only points to points, but also the collection of open sets of one space bijectively to the collection of open sets of another space. Without a clear study of open sets, there is no way to study homeomorphisms.

Homeomorphisms are a fundamental object of study, but many classification results need a proof that two topological spaces are not homeomorphic. The space of homeomorphisms of a given space can be a difficult object to understand, so showing the absence of a homeomorphism can be impossible. So instead, we extract properties of topological spaces that are invariant under homeomorphism. For instance, orientability is invariant under homeomorphism, so if one space is orientable and the other is not, these two spaces are different. Another frequently useful tool is invariance of domain (used to show that (chunks of) Euclidean spaces of different dimensionality are nonhomeomorphic).

An additional continuous map is the homotopy -- such a map continuously deforms a subset of a space into another subset. When people first (successfully) think of deforming a coffee cup into a doughnut, they think of a homotopy in the ambient $3$-dimensional Euclidean space. Homotopies lead to an invariant that has proven to be very useful: a space's set of homotopy groups. These are the groups of equivalence classes of $n$-spheres, equivalent under homotopy. (Notation: the group $H_n$ denotes the group if equivalence classes of $n$-spheres, $S^n$.) Consider a torus -- there are two equivalence classes of $1$-spheres (circles) that generate all paths on the torus up to homotopy. Taking the torus to be $S^1 \times S^1$, where $S^1$ is the closed interval $[0,1]$ under the quotient $0 \sim 1$, these two generators are homotopic to the loop $\alpha$, $\{0\}\times S^1$, and the loop $\beta$, $S^1 \times \{0\}$. These two loops intersect at $\{0\} \times \{0\}$, which we will call a basepoint for the first homotopy group, $H_1(S^1 \times S^1; \mathbb{Z})$. (The basepoint is a technical requirement -- all loops are taken to start at the basepoint and end there. It can be relaxed, but then one does not get a group and cannot use the extensive machinery developed for groups. The "$\mathbb{Z}$" there is the set of coefficients -- which we are about to use; other choices that integer coefficients are possible. The homotopy group $H_1$ is also called the fundamental group.) Note that we can make a loop that goes the long way around the torus twice, three times, et c. We can even go backwards around this loop as many times as we like. So each element of the $\mathbb{Z}$-modules $\mathbb{Z}\alpha$ and $\mathbb{Z}\beta$ correspond to distinct loops in this $H_1$. It is a standard proof in an algebraic topology class that $H_1$ of the circle is $\mathbb{Z}$, which we have just essentially observed. It is another standard result (usually an exercise) that the $H_1$ of a Cartesian product is a direct product of the fundamental groups of the spaces in the Cartesian product. After those exercises, it is clear that $H_1(S^1 \times S^1; \mathbb{Z}) \cong \mathbb{Z} \times \mathbb{Z}$, where one multiplicand corresponds to the number of times around $\alpha$ and the other to the number of times around $\beta$.

So now we can determine that many spaces are not the torus in disguise (not the torus after some convoluted and obfuscating homeomorphism). To be a torus, your fundamental group (with integer coefficients) must be abelian, rank $2$, and torsion-free. (This completely characterizes the torus.) Unfortunately, homotopy groups are not a strong enough invariant to distinguish all non-homeomorphic topological spaces. (This becomes unavoidably true for $4$-dimensional, and higher, manifolds. It is already true for $3$-manifolds, see lens spaces. And manifolds are not "most" topological spaces -- they're just the ones I study.)

I think I've addressed the bridge you asked about in your question, but a few words about knots, since you mentioned them explicitly... A knot is a loop in an ambient space, usually $S^3$, the one-point compactification (another idea from point-set topology) of $3$-dimensional Euclidean space. A knot is just an injection from $S^1$ into the space, so doesn't have very much structure itself. What is interesting is the space left after deleting the knot. The fundamental group of this complementary space is a knot invariant, so an ambient isotopy of the space taking the knot to another knot preserves this fundamental group. Unfortunately, there are inequivalent knots whose complements have the same fundamental group -- we need more invariants.

Answer 2

Topology helps you define continuity without using the notion of distance but rather by using the notion of what we call open sets (an element $T$ is what we call an open set).

Using the definition of topology you can also determine several properties of the space $X$ as well as properties of its subsets. An example of such a property is "connectedness". You can think of connectedness as something as holding in one piece

It's important to notice however that this notion of continuity can be very unintuitive since it relies heavily on the topology we define on a set (i.e. a function from one set $(X,T_X)$ to another $(Y,T_Y)$ can be continuous for some topologies $T_X$ and $T_Y$ and not continuous with other topologies !

Now you might want to study functions between topological spaces.

A function $ f : (X,T_X) \rightarrow (Y,T_Y) $ is called a homeomorphism if $f$ is a bijection, $f$ is continuous and $f^{-1}$ is continuous.

If there exists an homeomorphism between the space $X$ and $Y$ we say that those spaces are homeomorph

This kind of functions preserve properties i.e. if $X$ has a property then Y has the same property.

(very)Roughly speaking you can think of an homeomorphism as a continuous deformation of an object (no tearing or filling allowed !). If you take two geometric objects like a square and a circle then I'm sure you can imagine that you could bend the square so that it would become a circle. You can then say that the square and a circle are homeomorph.

A classical example of this is to say that a mug and a doughnut are topologically speaking identical you can stretch a mug into the shape of a doughnut just like in this animation. Notice how we can't stretch a doughnut into a ball because there is a hole in the middle of it ! Rolling the doughnut into a ball shape wouldn't make our transformation continuous ! this means holes in objects are properties that are carried over by homeomorphisms.

Hopes this helped a bit :)