Theorem 2.1(Heine Borel):

A subset of $\mathbb{R}^n$ is compact (with respect to the subspace topology) if and only if it is closed and bounded.

The theorem provides a profusion of examples of compact spaces.

1. The unit sphere $S^{n-1} = \{(x_1, x_2,\dots, x_n)\in \mathbb{R}^n \;\vert\; x_1^2 + x_2^2 + \dots + x_n^2 = 1\}$ is compact.
2. The unit square $I^2 = [0, 1]\times [0,1]$ is compact.
3. The set of all $3\times 3$ matrices is clearly homeomorphic to $\mathbb{R}^9$. Then the set of all $3\times 3$ orthogonal matrices, denoted by $O(3, \mathbb{R})$ is is compact. That is to say the orthogonal group is compact. The result readily generalizes to the group of $n\times n$ orthogonal matrices.
4. Think of the set of all $n\times n$ matrices with complex entries as $\mathbb{C}^{n^2}$ which in turn may be viewed as $\mathbb{R}^{2n^2}$. The set of all $n\times n$ unitary matrices is then easily seen to be a compact space. These matrices form a group known as the unitary group $U(n)$.
5. The set of all $n\times n$ unitary matrices with determinant one is also a closed bounded subset of $\mathbb{C}^{n^2}$ and so is compact. This is the special unitary group $SU(n)$.