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Let $\ell^\infty(\mathbb{N})$ denote the set of bounded real sequences $(a_n)_{n\in\mathbb{N}}$. The $\lim$ operator is a partial linear operator from $\ell^\infty(\mathbb{N})$ to $\mathbb{R}$. With t …

If $X, Y$ are topological spaces such that for every continuous map $f: X\to Y$ and any $K\subseteq Y$ compact, $f^{-1}(K)$ is a compact of subset of $X$, then every continuous function is measurable. …

In the answer to the following comment, it is shown why $\rho(x,x') = 0$ implies $x = x'$: https://math.stackexchange.com/questions/163678/a-proof-that-skorohod-metric-is-a-metric

The answer to the question is No. Let $x:[0,1] \to [0,1]$ be defined by $x(t) = 0$ for $t\in [0,1/2[$ and $x(t) = 1$ for $t\in[1/2, 1]$. Clearly, $x$ is cadlag and therefore a member of $D([0,1], R)$. …

One important class of spaces for which the two definitions mentioned in the post are equivalent, are the first-countable spaces. One of the most important properties of any first-countable space $(X, …