First, note that a mapping between metric spaces is continuous if and only if the inverse image of an open set is always open. There are various concepts for metric spaces that you can likewise find equivalent formulations for in terms of open (and closed) sets, for example compactness. Convergence of a sequence to a point can be rephrased in terms of neighbourhoods of the point, with no reference to any ε. Then you could, for example, notice how you can talk about pointwise convergence of functions, but there is no corresponding metric. So you need a more general framework for talking about different kinds of convergence, and soon enough, topological spaces won't seem so strange anymore.
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