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A second-countable space is said to satisfy the second axiom of countability. Like other countability axioms, the property of being second-countable restricts the number of open sets that a space can have. Many "well-behaved" spaces in mathematics are second-countable. For example, Euclidean space (R n) with
A countable product of second-countable spaces is second countable, but an uncountable product of second-countable spaces need not even be first countable. We can construct an example of a separable topological space that is not second countable. Consider any uncountable set , pick some , and define the topology to be the collection of all sets ...
second-countable space: the topology has a countable base; separable space: there exists a countable dense subspace; Lindelöf space: every open cover has a countable subcover; σ-compact space: there exists a countable cover by compact spaces; Relations: Every first countable space is sequential. Every second-countable space is first-countable ...
For example, there are many compact spaces that are not second-countable. A metric space is Lindelöf if and only if it is separable, and if and only if it is second-countable. [6] Every regular Lindelöf space is normal. [7] Every regular Lindelöf space is paracompact. [8] A countable union of Lindelöf subspaces of a topological space is ...
sequential space: a set is open if every sequence convergent to a point in the set is eventually in the set; first-countable space: every point has a countable neighbourhood basis (local base) second-countable space: the topology has a countable base; separable space: there exists a countable dense subset
For example, second countability and metrisability are hereditary properties. Sequentiality and Hausdorff compactness are weakly hereditary, but not hereditary. [1] Connectivity is not weakly hereditary. If P is a property of a topological space X and every subspace also has property P, then X is said to be "hereditarily P".
A second-countable space is one that has a countable base. The Zariski topology on the spectrum of a ring has a base consisting of open sets that have specific useful properties. For the usual base for this topology, every finite intersection of basic open sets is a basic open set.
For any manifold the properties of being second-countable, Lindelöf, and σ-compact are all equivalent. Every second-countable manifold is paracompact, but not vice versa. However, the converse is nearly true: a paracompact manifold is second-countable if and only if it has a countable number of connected components. In particular, a connected ...