Topological spaces and open sets

Question

• “Also in $\mathcal{T}$” can be denoted as either subset or element inclusion?
• How to understand the topology induced by a norm (correct phrasing)?
  • What about other norms?
• Why is a topological space discrete $⟺$ every singleton is open? Isn’t this by definition—the fact that the discrete topology includes all the open singletons? Is there such a thing as a non-discrete topology with non-open singletons?
• Why, in general, do we require finite intersections, aside from the intuition that an open set should have something like a neighborhood for each point?

---

Overview §

Definition: Topology, topological space

A topology on a set $X$ is a collection $\mathcal{T}$ of subsets of $X$ (i.e., a subset of the power set $\mathcal{T}\subseteq \mathcal{P}(X)$) that has the following properties:

1. $\varnothing$ and $X$ are in $\mathcal{T}$;
2. The union of the elements of any (arbitrary) sub-collection of $\mathcal{T}$ is also in $\mathcal{T}$;
3. The intersection of the elements of any finite sub-collection of $\mathcal{T}$ is also in $\mathcal{T}$.

A topological space is an ordered pair $(X,\mathcal{T})$, where $X$ is a set and $\mathcal{T}$ is a topology on $X$.

Two points are topologically indistinguishable if they cannot be separated by open sets; the Hausdorff condition ensures that any two distinct points in a topological space are distinguishable.

Topologies can be compared by the granularity of their sets.

Definition: Finer, coarser, and comparable topologies

Two topologies $\mathcal{T},\mathcal{T}’$ on a set $X$ are comparable if either $\mathcal{T}\subseteq \mathcal{T}’$ or $\mathcal{T}’\subseteq \mathcal{T}$. We say $\mathcal{T}’$ is finer than $\mathcal{T}$ if $\mathcal{T}’$ has more open sets than $\mathcal{T}$; that is, if $\mathcal{T}\subseteq \mathcal{T}’$. In this case, we also say $\mathcal{T}$ is coarser than $\mathcal{T}’$.

---

Open sets in topological space §

Elements of $\mathcal{T}$ are called open sets of $X$. Recall that a subset $U\subseteq {\mathbb{R}}^n$ is called open if for all $x\in U$, there exists some (sufficiently small) $ϵ>0$ such that the $ϵ$-ball about $x$

$$B_{ϵ}(x)=\{y\in {\mathbb{R}}^n||x-y|<ϵ\}$$

is completely contained in $U$.

Definition: Open set in a topological space

If $X$ is a topological set with topology $\mathcal{T}$, we say a subset $U\subset X$ is an open set of $X$ if $U\in \mathcal{T}$.

Thus, we can also say that a topological space is a set $X$ together with a collection of open subsets of $X$ such that $\varnothing$ and $X$ are both open, and arbitrary unions and finite intersections of open subsets are open.

---

Examples §

The motivating example §

The standard, or Euclidean, norm $\Vert \cdot {\Vert }_2$ in ${\mathbb{R}}^n$ defined by

$$\Vert x{\Vert }_2=\sqrt{x_1^2+\cdots +x_n^2}$$

gives the distance between two points as $\Vert x-y{\Vert }_2$. The topology induced by this norm is precisely the collection of open balls $B_{ϵ}(x)$, where $ϵ>0$ is some real number and $x\in {\mathbb{R}}^n$.

The arbitrary set $X=\{a,b,c\}$ and its non-examples §

An arbitrary three-element set $X=\{a,b,c\}$ has 9 unique topologies up to permutation.

The following collections of subsets are not topologies (draw them out!):

• $\{\varnothing ,X,\{a\},\{b\}\}$
• $\{\varnothing ,X,\{a,b\},\{b,c\}\}$

Discrete and indiscrete topologies §

Transclude of Discrete-topology#^aa370a

Finite complement topology §

Definition: Finite complement topology

The finite complement topology of a set $X$ is the collection of all subsets $U\subseteq X$ such that the complement $X\backslash U$ is either finite or all of $X$. We write $X_f$ to denote the set $X$ equipped with the finite complement topology.

Link to original

Subspace (induced) topology §

Definition: Subspace (induced) topology

Given a topological space $(X,\mathcal{T})$ and a subset $A\subseteq X$, the subspace or induced topology on $A$ is defined by the collection

$${\mathcal{T}}_A=\{A\cap U|U\in \mathcal{T}\}.$$

In other words, open sets in $A$ (i.e., a subset of $A$ that is open in the subset topology) are obtained by intersecting $A$ with open sets of $X$ (i.e., an open set in $(X,\mathcal{T}$)).

Link to original

Lower limit topology on $\mathbb{R}$ §

#wip

• Basis is the set of intervals [a, b)

---

Review §

Definitions §

• Topology, topological space
• Finer and coarser topologies
• Discrete and indiscrete topology
• Subspace (induced) topology

Exercises §

• Prove that the collection of all open sets is a topology on ${\mathbb{R}}^n$ called the standard topology. ⭐
• Given an arbitrary set $X=\{a,b,c\}$, give an example and a non-example of a topology.
• Prove that the following examples satisfy all the properties of a topology:
  • Discrete and indiscrete topologies
  • Finite topology (hint: for (T2) and (T3), use the fact that the complement of an intersection of a family of indexed sets is same as a union of complements between each indexed set and the ambient set, as well as the reverse; see Algebra of sets)
  • Subspace topology
• Show that $\mathbb{Z}\subseteq \mathbb{R}$ and $K:=\{1/n|n\in {\mathbb{Z}}_{>0}\}$ with the respective subspace topologies are discrete, but $K\cup \{0\}$ is not. ⭐