Open Set in Modified Metric Space

Determine if the interval $[1, 3)$ is an open set in the metric space of real numbers defined as $x = \{1 \leq x < 3\} \cup \{x > 4\}$, where the metric is the standard distance function.

Determining whether a set is open in a given metric space is a fundamental question in topology and real analysis. In this problem, we are considering the intervals on the real line with a specific metric and set definition. The core idea is to understand the definition of open sets within any metric space, where a set is deemed open if around every point in the set, you can find a neighborhood entirely contained within that set. This revolves around grasping the concept of neighborhoods in metric spaces, which in simpler terms, can be thought of as open balls centered at points in your set. Since the metric we're working with is the standard distance function, we need to determine if every point in our specified interval $[1, 3)$ can be enclosed in such a neighborhood without touching the boundary in terms of metric distance.

Furthermore, the problem introduces a non-standard set definition, adding the points greater than 4, which could potentially affect how we view the openness of a set in this particular space. Understanding how these disjoint unioned sets interact under the standard metric is crucial. This problem taps into the more general concept of topology within metric spaces, encouraging you to consider how merging intervals or point sets affects topological properties like openness and how metric definitions influence these properties.