You should know the following theorems from calculus on $\Bbb{R}$:
Extreme Value Theorem: A continuous function $f$ on a closed bounded interval attains its maximum and minimum value on that interval.
A continuous function on a closed bounded interval is uniformly continuous
Intermediate Value Theorem: Let $f$ be a continuous function on $\Bbb{R}$ such that there are real numbers $a,b$ for which $f(a) < 0$ and $f(b) > 0$. Then there exists $c \in (a,b)$ such that $f(c)= 0$.
The first two theorems make use of a topological property known as compactness. The Heine - Borel Theorem is what tells you that in $\Bbb{R}$ with Euclidean metric, being closed and bounded is equivalent to being compact. Number 3 makes use of the topological property that $\Bbb{R}$ with the euclidean metric is connected.
Number 2 is especially important to know why a continuous function on a closed bounded interval is Riemann Integrable; this should be enough motivation for you to study topology!
The following I think is a cool example of where things get gnarly in topology. In a metric space (in fact in a topological space that is at least $T_2$) this cannot happen but:
Put the trivial topology on $\Bbb{R}$, the family $A_n = (0,\frac{1}{n})$ is a countable collection of non-empty compact sets such that the intersection of any finite sub-collection is non-empty, but clearly
$\bigcap_{n=1}^\infty A_n = \emptyset .$