Uniform limit theorem

Empty set and whole space: ∅ , X ∈ T \varnothing,X\in\mathcal T ∅ , X ∈ T .
Arbitrary unions: the union of any collection of members of T \mathcal T T belongs to T \mathcal T T .
Finite intersections: the intersection of finitely many members of T \mathcal T T belongs to T \mathcal T T .

Uniform limit theorem. Let $X$ be a topological space and $Y$ a metric space. If $f_n:X\to Y$ are continuous functions and $f_n\to f$ uniformly, then $f:X\to Y$ is continuous. Uniform convergence means that for every $\varepsilon>0$ there exists $N$ such that

n\ge N\quad\Longrightarrow\quad d_Y\bigl(f_n(x),f(x)\bigr)<\varepsilon \quad\text{for every }x\in X.

Counterexample for pointwise convergence

$f_n(x) = x^n$ on $[0, 1]$ converges pointwise to a discontinuous limit:

f(x) = \begin{cases} 0 & x \in [0,1) \\ 1 & x = 1 \end{cases}.

The convergence is not uniform.