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In topology, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable local base. That is, for each x ∈ X there exists a sequence U1, U2, … of open neighborhoods of x such that for any neighborhood V there exists an integer i with Ui ⊆ V. Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ...
Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...
In mathematics, an axiom of countability is a property of certain mathematical objects (usually in a category) that requires the existence of a countable set with certain properties, while without it such sets might not exist. ...
In mathematics the term countable set is used to describe the size of a set, e. ...
This is a glossary of some terms used in the branch of mathematics known as topology. ...
This is a glossary of some terms used in the branch of mathematics known as topology. ...
Examples and counterexamples The majority of 'everyday' spaces in mathematics are first-countable. In particular, every metric space is first-countable. To see this, note that the set of open balls centered at x with radius 1/n for integers n > 0 form a countable local base at x. Euclid, detail from The School of Athens by Raphael. ...
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...
A synonym for ball (in geometry or topology, and in any dimension) is disk (or disc Geometry In metric geometry, a ball is a set containing all points within a specified distance of a given point. ...
An example of a space which is not first-countable is the cofinite topology on an uncountable set (such as the real line). In mathematics, a cofinite subset of a set X is a subset Y whose complement in X is a finite set. ...
In mathematics, the real line is simply the set of real numbers. ...
Another counterexample is the ordinal space [0,ω1] where ω1 is the smallest uncountable ordinal number. The element ω1 is a limit point of the subset [0,ω1) even though no sequence of elements in [0,ω1) has the element ω1 as its limit. In particular, ω1 does not have a countable local base. The subspace [0,ω1) is first-countable however, since ω1 is the only such point. In mathematics, an uncountable set is a set which is not countable. ...
Commonly, ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ...
In mathematics, informally speaking, a limit point (or cluster point) of a set S in a topological space X is a point x in X that can be approximated by points of S other than x as well as one pleases. ...
Properties One of the most important properties of first-countable spaces is that given a subset A, a point x lies in the closure of A if and only if there exists a sequence {xn} in A which converges to x. This has consequences for limits and continuity. In particular, if f is a function on a first-countable space, then f has a limit L at the point x if and only if for every sequence xn → x, where xn ≠ x for all n, we have f(xn) → L. Also, if f is a function on a first-countable space, then f is continuous if and only if whenever xn → x, then f(xn) → f(x). In mathematics, the closure of a set S consists of all points which are intuitively close to S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior. ...
In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ...
Limit of a sequence is one of the oldest concepts in mathematical analysis. ...
In mathematics, the limit of a function is a fundamental concept in mathematical analysis. ...
In topology, a continuous function is generally defined as one for which preimages of open sets are open. ...
In first-countable spaces, sequential compactness and countable compactness are equivalent properties. However, there exist examples of sequentially compact, first-countable spaces which aren't compact (these are necessarily non-metric spaces). One such space is the ordinal space [0,ω1). Every first-countable space is compactly generated. In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ...
In topology, a compactly generated space is a topological space X satisfying the following condition: a subspace A is closed in X if and only if A ∩ K is closed in K for all compact subspaces K ⊆ X. Equivalently, one can replace closed with open in this definition. ...
Every subspace of a first-countable space is first-countable. Any countable product of a first-countable space is first-countable, although uncountable products need not be. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...
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