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Wolfram defines an open set as

A set for which every point in the set has a neighborhood lying in the set.

Is my understanding correct?: an open set has an infinite number of points because no matter how close you get to the boundary, you still have a neighborhood, while a closed set has only a finite number of points?

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    a) The empty set is open. b) It depends on the space, if we're talking about $\mathbb{R}^n$ with the standard topology, then yes, every non-empty open set is infinite (has cardinality $2^{\aleph_0}$ even). If we're talking about $\mathbb{Z}$ with its standard topology, then every set is open, so a non-empty open set may be finite or infinite. – 2017-02-04
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    In the discrete topology , **anything** is open (and also closed), so any subset of $\;\Bbb R\;$ , with this topology, is open. You now can see there are manu *finite* subsets and these are open. – 2017-02-04
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    Also, certainly it is not true that a closed set is finite! Any closed interval is finite, for instance. Any open set is contained in a closed set: its closure (or just the entire underlying space). – 2017-02-04

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It depends on the topology. If you have a set with the discrete topology, that statement is not true. In the discrete topology, all subsets are open, so any point is an open set, and it doesn't have an infinite number of points.

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No your conclusion is false. Consider for example the discrete topology on a set $X$ finite or infinite. Then $\{x\}$ is an open set.

Now, take the usual topology on $\mathbb{R}$ and the closed ball $B_\epsilon(x)$ for $x\in\mathbb{R}$. Then $B_\epsilon(x)$ has an infinite number of points.

In other words, cardinality and openess/closedness of sets are not directly correlated.