Talk:Q29654355

Closed interval?

@Infovarius: A closed interval is an interval containing both its endpoints, so that it can be written of the form $[a,b]$. In the real line (or any linearly ordered topological space), every closed interval is a closed set but there are intervals that are closed sets but not closed intervals, such as the whole real line, $[0,\infty )$, etc. 慈居 (talk) 23:15, 22 August 2023 (UTC)

Really? It's about endpoints, not limit points? --Infovarius (talk) 18:09, 24 August 2023 (UTC)

I don't think the definition using "limit points" is an acceptable definition. Actually, the definition of limit points depends on the choice of topology on the ordered set. So it does not apply to the intervals ordered sets without confusion. There are many topologies that an ordered set can wear, e.g. the interval topology, the upper/lower topology, the Scott/Lawson topology, the Sorgenfrey topology, etc. Limit points of an interval in an ordered set need not be endpoints. For example, in $(0,1)\subseteq \mathbb {R}$, the limit points are the points in $[0,1]$. Conversely, endpoints need not be limit points. Consider the ordered set $S=\{0,1,2\}$ with order $0<1<2$ and with the interval topology. The interval $(0,2)_{S}=\{1\}$ has no limit points, hence it is a "closed interval" in the "limit point" sense. 慈居 (talk) 23:32, 24 August 2023 (UTC)

In the example above, $(0,2)_{S}=[1,1]$, so it may not look that problematic. Then let us consider the ordered set $T=\{0\}\cup (1,2)\cup \{3\}$. In $T$, $(0,3)_{T}$ is a "closed interval" in the limit point sense, but it can not be written of the form $[a,b]$, because the interval has no least or greatest elements. 慈居 (talk) 23:37, 24 August 2023 (UTC)

Please see [1], if you want a reference. 慈居 (talk) 00:42, 25 August 2023 (UTC)