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Also, if is a function on a first-countable space, then is continuous if and only if whenever , then () (). In first-countable spaces, sequential compactness and countable compactness are equivalent properties. However, there exist examples of sequentially compact, first-countable spaces that are not compact (these are necessarily not ...
If X is a first-countable space and countable choice holds, then the converse also holds: any function preserving sequential limits is continuous. In particular, if X is a metric space, sequential continuity and continuity are equivalent.
In mathematics and statistics, a quantitative variable may be continuous or discrete if it is typically obtained by measuring or counting, respectively. [1] If it can take on two particular real values such that it can also take on all real values between them (including values that are arbitrarily or infinitesimally close together), the variable is continuous in that interval. [2]
Every function whose domain is a topological space and codomain X is continuous. X is path-connected and so connected. X is second-countable, and therefore is first-countable, separable and Lindelöf. All subspaces of X have the trivial topology. All quotient spaces of X have the trivial topology
As a quotient of a metric space, the result is sequential, but it is not first countable. Every first-countable space is Fréchet–Urysohn and every Fréchet-Urysohn space is sequential. Thus every metrizable or pseudometrizable space — in particular, every second-countable space, metric space, or discrete space — is sequential.
The example of the set of all real numbers with the standard topology shows that neither local compactness nor σ-compactness nor paracompactness imply countable compactness. Closed subspaces of a countably compact space are countably compact. [8] The continuous image of a countably compact space is countably compact. [9]
This includes first countable spaces, Alexandrov-discrete spaces, finite spaces. Every CG-3 space is a T 1 space (because given a singleton { x } ⊆ X , {\displaystyle \{x\}\subseteq X,} its intersection with every compact Hausdorff subspace K ⊆ X {\displaystyle K\subseteq X} is the empty set or a single point, which is closed in K ...
In particular, every continuous function on a separable space whose image is a subset of a Hausdorff space is determined by its values on the countable dense subset. Contrast separability with the related notion of second countability , which is in general stronger but equivalent on the class of metrizable spaces.