Difference between Open sets of Interval and Open Sets of Topological Space

Question

I am trying to understand the difference between open sets on a real line and open sets in a topological space. For example, while reading about open sets in Real line, it says:

Recall the following definitions about open and closed sets in $\mathbb{R}^d$.

Open Sets: Write

$B_d(x,r) :=\{y \in R^d: |y-x| < r\}$

for the open ball of radius $r$ about $x \in \mathbb{R}^d$.

A set $G \subset \mathbb{R}^d$ is open if for all $x \in G$ there exists an $r > 0$ such that $B(x,r) \subset G$.

Now if talk about topological space:

A topological space, also called an abstract topological space, is a set $X$ together with a collection of open subsets $T$ that satisfies the four conditions:

1. The empty set $\emptyset$ 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 $T$.

Members of the $T$ are called open sets

Now, how these open sets of real line and Topological space are related?

Answer 1

The real line is a main example of a topological space. If you define $$ \mathcal T = \{ A\subseteq\mathbb R \mid A\text{ satisfies the }{\it first}\text{ definition of open set}\}$$ then you can (fairly easily) prove that this particular $\mathcal T$ satisfies your definition of topology.

Therefore as long as you're speaking about that topology (the "standard topology" on $\mathbb R$), saying that members of $\mathcal T$ are called "open" does agree with your initial interval-based definition of "open".

However, the point of defining "topology" abstractly is that you can now talk about other topologies where "open" does not have anything in particular to do with intervals.

Answer 2

The first is a special case of a topological space; whereas the second part describes topological spaces in general...

$\mathbb R^d$ is in fact a metric space, so that we can talk about open balls of given radius. (I'm referring here to the standard Euclidean metric, though there are others). In any metric space the open balls form a basis for a topology. That is what is being described above.

This isn't the case in every topological space... All we need for a topological space generally is a collection of sets which are defined to be open, and which satisfy the conditions given (in the second part)...

Answer 3

Any open set in $\mathbb R$ is a union of open intervals; one says that the open intervals form a basis for the topology of $\mathbb R$. Similarly, open balls form a basis for the topology of $\mathbb R^n$.

Answer 4

The topology of $\mathbb {R^n}$ is a special case of general topology.

The open sets in $\mathbb {R^n}$ satisfy the four axioms of the open sets in General topology.

The topology of $\mathbb {R^n}$ is special topology called Metric Topology in which distances are defined while in general topology the notion of distance is not part of the picture.