We start our study of topology by looking at real analysis, and seeing what motivates our definitions in topology.
Motivation
Much of the study of point-set topology is motivated by abstracting ideas from Real Analysis. In particular, we get our definition of a ‘topology’ from studying open intervals in \mathbb{R}.
When a,b \in \mathbb{R}, and a < b, we use the following notations for intervals
An open interval is defined
(a,b) = \{x \in \mathbb{R} \mid a < x < b\}
A half open interval is defined as either
(a,b] = \{x \in \mathbb{R} \mid a < x \leq b\}\\
[a,b) = \{x \in \mathbb{R} \mid a \leq x < b\}
And a closed interval is defined
[a,b] = \{x \in \mathbb{R} \mid a \leq x \leq b\}
Open intervals don’t contain their endpoints, which leads to interesting properties.
Let x be some point in the open interval (a,b). There there exists some \delta > 0 such that (x - \delta, x + \delta) \subset (a,b).
This means for each point x in the interval (a,b), you can construct an interval around x that is completely contained in (a,b).
TO-DO: Diagram
Let x be some point in the open interval (a,b). Then define delta to be half of the minimum distance from x to the endpoints of the interval;
\delta = \frac{\min\{|a-x|,|b-x|\}}{2}.
Then one can clearly see x + \delta < a, and x - \delta > b. Thus (x - \delta,x+\delta) \subset (a,b).
Open sets
This property of open intervals is much more important than it looks, and it can be applied to other types of sets.
Let U be a set, and x \in U. Then x is an interior point of S if there exists some \delta > 0 such that (x - \delta, x + \delta) \subset U.
Let [0,4] be an interval. Then 2 is an interior point of [0,4]. We cab check this by letting \delta equal 1. Then
(2 - 1, 2 + 1) = (1,3) \subset [0,4]
With this definition, we can abstract our previous property of open intervals to other sets.
A set U is open if every point x \in U is an interior point.