Correct option is C
A connected metric space cannot be decomposed into two or more disjointnon-empty open subsets.
This implies that if the space is connected and has atleast two points,
it must :
1. Have infinitely many points.
2. Be uncountable (because a countableconnected space would violate the properties of being a connected metric space).
Thus: - X cannot have finitely many points because a finite metric spacecannot be connected
unless it has only one point. - X cannot have countablymany points but not be finite because
such a set cannot satisfy the conditionsof connectedness in a metric space.
Therefore, the only possibility is that X has uncountably many points.
Correct Answer : Option (C).
