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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 ...
For non first-countable spaces, sequential continuity might be strictly weaker than continuity. (The spaces for which the two properties are equivalent are called sequential spaces.) This motivates the consideration of nets instead of sequences in general topological spaces.
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.
Every second-countable space is first-countable, separable, and Lindelöf. Semilocally simply connected A space X is semilocally simply connected if, for every point x in X, there is a neighbourhood U of x such that every loop at x in U is homotopic in X to the constant loop x. Every simply connected space and every locally simply connected ...
A space is countably compact if every countable open cover has a finite subcover. Pseudocompact. A space is pseudocompact if every continuous real-valued function on the space is bounded. σ-compact. A space is σ-compact if it is the union of countably many compact subsets. Lindelöf. A space is Lindelöf if every open cover has a countable ...
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
For non-first-countable spaces, sequential continuity might be strictly weaker than continuity. (The spaces for which the two properties are equivalent are called sequential spaces.) This motivates the consideration of nets instead of sequences in general topological spaces.
Every discrete space is first-countable; it is moreover second-countable if and only if it is countable. Every discrete space is totally disconnected. Every non-empty discrete space is second category. Any two discrete spaces with the same cardinality are homeomorphic. Every discrete space is metrizable (by the discrete metric).