  Given a metric space $\mathcal{M}=(X,\delta)$, a weighted graph $G$ over $X$
is a metric $t$-spanner of $\mathcal{M}$ if for every $u,v \in X$,
$\delta(u,v)\le d_G(u,v)\le t\cdot \delta(u,v)$, where $d_G$ is the shortest
path metric in $G$. In this paper, we construct spanners for finite sets in
metric spaces in the online setting. Here, we are given a sequence of points
$(s_1, \ldots, s_n)$, where the points are presented one at a time (i.e., after
$i$ steps, we saw $S_i = \{s_1, \ldots , s_i\}$). The algorithm is allowed to
add edges to the spanner when a new point arrives, however, it is not allowed
to remove any edge from the spanner. The goal is to maintain a $t$-spanner
$G_i$ for $S_i$ for all $i$, while minimizing the number of edges, and their
total weight.
  We construct online $(1+\varepsilon)$-spanners in Euclidean $d$-space,
$(2k-1)(1+\varepsilon)$-spanners for general metrics, and
$(2+\varepsilon)$-spanners for ultrametrics. Most notably, in Euclidean plane,
we construct a $(1+\varepsilon)$-spanner with competitive ratio
$O(\varepsilon^{-3/2}\log\varepsilon^{-1}\log n)$, bypassing the classic lower
bound $\Omega(\varepsilon^{-2})$ for lightness, which compares the weight of
the spanner, to that of the MST.
