Teleparallelism: Difference between revisions
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In [[topology]], a branch of [[mathematics]], 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]] [[neighbourhood system|neighbourhood basis]] (local base). That is, for each point ''x'' in ''X'' there exists a [[sequence]] ''N''<sub>1</sub>, ''N''<sub>2</sub>, … of [[neighbourhood (topology)|neighbourhoods]] of ''x'' such that for any neighbourhood ''N'' of ''x'' there exists an integer ''i'' with ''N''<sub>''i''</sub> [[subset|contained in]] ''N''. | |||
Since every neighborhood of any point is contained in an open neighborhood of that point the [[neighbourhood system|neighbourhood basis]] can be chosen w.l.o.g. to consist of open neighborhoods. | |||
==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 ball]]s centered at ''x'' with radius 1/''n'' for integers ''n'' > 0 form a countable local base at ''x''. | |||
An example of a space which is not first-countable is the [[cofinite topology]] on an uncountable set (such as the [[real line]]). | |||
Another counterexample is the [[ordinal space]] ω<sub>1</sub>+1 = [0,ω<sub>1</sub>] where ω<sub>1</sub> is the [[first uncountable ordinal]] number. The element ω<sub>1</sub> is a [[limit point]] of the subset <nowiki>[</nowiki>0,ω<sub>1</sub>) even though no sequence of elements in <nowiki>[</nowiki>0,ω<sub>1</sub>) has the element ω<sub>1</sub> as its limit. In particular, the point ω<sub>1</sub> in the space ω<sub>1</sub>+1 = [0,ω<sub>1</sub>] does not have a countable local base. Since ω<sub>1</sub> is the only such point, however, the subspace ω<sub>1</sub> = <nowiki>[</nowiki>0,ω<sub>1</sub>) is first-countable. | |||
The [[quotient space]] <math>\mathbb{R}/\mathbb{N}</math> where the natural numbers on the real line are identified as a single point is not first countable. However, this space has the property that for any subset A and every element x in the closure of A, there is a sequence in A converging to x. A space with this sequence property is sometimes called a [[Sequential space#Fréchet–Urysohn space|Fréchet-Urysohn space]]. | |||
First-countability is strictly weaker than [[second-countability]]. Every [[second-countable space]] is first-countable, but any uncountable [[discrete space]] is first-countable but not second-countable. | |||
==Properties== | |||
One of the most important properties of first-countable spaces is that given a subset ''A'', a point ''x'' lies in the [[closure (topology)|closure]] of ''A'' if and only if there exists a [[sequence]] {''x''<sub>''n''</sub>} in ''A'' which [[limit of a sequence|converges]] to ''x''. This has consequences for [[limit of a function|limits]] and [[continuity (topology)|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 ''x''<sub>''n''</sub> → ''x'', where ''x''<sub>''n''</sub> ≠ ''x'' for all ''n'', we have ''f''(''x''<sub>''n''</sub>) → ''L''. Also, if ''f'' is a function on a first-countable space, then ''f'' is continuous if and only if whenever ''x''<sub>''n''</sub> → ''x'', then ''f''(''x''<sub>''n''</sub>) → ''f''(''x''). | |||
In first-countable spaces, [[sequentially compact space|sequential compactness]] and [[countably compact space|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 [[Order topology|ordinal space]] <nowiki>[</nowiki>0,ω<sub>1</sub>). Every first-countable space is [[compactly generated space|compactly generated]]. | |||
Every [[subspace (topology)|subspace]] of a first-countable space is first-countable. Any countable [[product space|product]] of a first-countable space is first-countable, although uncountable products need not be. | |||
==See also== | |||
*[[Second-countable space]] | |||
*[[Separable space]] | |||
==References== | |||
*{{Springer|id=f/f040430|title=first axiom of countability}} | |||
{{DEFAULTSORT:First-Countable Space}} | |||
[[Category:General topology]] | |||
[[Category:Properties of topological spaces]] | |||
Revision as of 17:20, 30 January 2014
In topology, a branch of mathematics, 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 neighbourhood basis (local base). That is, for each point x in X there exists a sequence N1, N2, … of neighbourhoods of x such that for any neighbourhood N of x there exists an integer i with Ni contained in N. Since every neighborhood of any point is contained in an open neighborhood of that point the neighbourhood basis can be chosen w.l.o.g. to consist of open neighborhoods.
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.
An example of a space which is not first-countable is the cofinite topology on an uncountable set (such as the real line).
Another counterexample is the ordinal space ω1+1 = [0,ω1] where ω1 is the first 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, the point ω1 in the space ω1+1 = [0,ω1] does not have a countable local base. Since ω1 is the only such point, however, the subspace ω1 = [0,ω1) is first-countable.
The quotient space where the natural numbers on the real line are identified as a single point is not first countable. However, this space has the property that for any subset A and every element x in the closure of A, there is a sequence in A converging to x. A space with this sequence property is sometimes called a Fréchet-Urysohn space.
First-countability is strictly weaker than second-countability. Every second-countable space is first-countable, but any uncountable discrete space is first-countable but not second-countable.
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 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.
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.
See also
References
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