Standard Topology
A SOFT BALL ABOUT EVERY POINT
A topology can simply be declared: someone says which subsets are the open sets. On \(\mathbb{R}^d\) the usual choice is not declared that way. A set is open when every one of its points has some ball about it lying inside the set.
The definition
For \(p \in \mathbb{R}^d\) and \(r \in \mathbb{R}^+\), the soft ball of radius \(r\) about \(p\) is:
\[ B_r(p) := \Big\{\, (q_1, \dots, q_d) \in \mathbb{R}^d \;\Big|\; \textstyle\sum_{i=1}^{d} (q_i - p_i)^2 < r^2 \,\Big\}. \]
The standard topology on \(\mathbb{R}^d\) selects from the powerset of \(\mathbb{R}^d\):
\[ \mathcal{O}_s := \big\{\, U \in \mathcal{P}(\mathbb{R}^d) \;\big|\; \forall\, p \in U \;\; \exists\, r \in \mathbb{R}^+ : B_r(p) \subseteq U \,\big\}. \]
The radius depends on the point
\(p\) is chosen before \(r\), so each point may pick its own radius, and in general it must: a point far inside a set can afford a large ball, a point near the edge only a small one.
The balls are themselves open
Write \(\rho(q,p) := \sqrt{\sum_{i=1}^{d} (q_i - p_i)^2}\), so that \(q \in B_r(p)\) says exactly \(\rho(q,p) < r\). Take \(q \in B_r(p)\) and let \(s := r - \rho(q,p) > 0\). Any \(x \in B_s(q)\) satisfies \(\rho(x,p) \leq \rho(x,q) + \rho(q,p) < s + \rho(q,p) = r\), so \(B_s(q) \subseteq B_r(p)\). Had the inequality been \(\leq\), this would fail at exactly the points on the sphere, where \(\rho(q,p) = r\) gives \(s = 0\) and no positive radius is left.
Checking a set by hand
On \(\mathbb{R}\), \(B_r(p)\) is the interval \((p-r,\, p+r)\). Then \((0,1)\) is an open set of \(\mathbb{R}\): for \(p\) in it take \(r = \min(p,\, 1-p)\). And \([0,1)\) is not, since \(0\) belongs to it and every \((-r,\, r)\) holds negative numbers.
Why finite intersections and arbitrary unions
Condition (iii) of a topology holds automatically: a point of the union lies in some member, and that member's radius serves. A pairwise intersection takes the smaller of two radii, which survives any finite number of repetitions. It does not survive an infinite family: \(\bigcap_n \left(-\tfrac{1}{n}, \tfrac{1}{n}\right) = \{0\}\), where the radii shrink to nothing and the result is not open.