We study dynamical systems that have bounded complexity with respect to three kinds metrics: the Bowen metric $d_{n}$, the max-mean metric $\hat{d}_{n}$ and the mean metric $\bar{d}_{n}$, both in topological dynamics and ergodic theory. It is shown that a topological dynamical system $(X,T)$ has bounded complexity with respect to $d_{n}$ (respectively $\hat{d}_{n}$) if and only if it is equicontinuous (respectively equicontinuous in the mean). However, we construct minimal systems that have bounded complexity with respect to $\bar{d}_{n}$ but that are not equicontinuous in the mean. It turns out that an invariant measure $\unicode[STIX]{x1D707}$ on $(X,T)$ has bounded complexity with respect to $d_{n}$ if and only if $(X,T)$ is $\unicode[STIX]{x1D707}$-equicontinuous. Meanwhile, it is shown that $\unicode[STIX]{x1D707}$ has bounded complexity with respect to $\hat{d}_{n}$ if and only if $\unicode[STIX]{x1D707}$ has bounded complexity with respect to $\bar{d}_{n}$, if and only if $(X,T)$ is $\unicode[STIX]{x1D707}$-mean equicontinuous and if and only if it has discrete spectrum.

Let C(X, Y) denote the set of continuous functions from a metric space X to a metric space Y. Viewing elements of C(X, Y) as closed subsets of X × Y, we say {fn} converges topologically to f if Li fn = Lsfn = f. If X is connected, then topological convergence in C(X,R) does not imply pointwise convergence, but if X is locally connected and Y is locally compact, then topological convergence in C(X, Y) is equivalent to uniform convergence on compact subsets of X. Pathological aspects of topological convergence for seemingly nice spaces are also presented, along with a positive Baire category result.