Examples of 'metrizable' in a sentence
Meaning of "metrizable"
Metrizable is a mathematical term used to describe a topological space that can be equipped with a metric that induces the given topology
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- measurable, quantifiable
- Of a topological space: for which a metric exists that will induce the original topology.
How to use "metrizable" in a sentence
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metrizable
A finite space is metrizable only if it is discrete.
The weak topology is always metrizable.
Completely metrizable spaces are often called topologically complete.
All manifolds are metrizable.
A manifold is metrizable if and only if it is paracompact.
A topological space whose topology can be described by a metric is called metrizable.
A space is metrizable if it is homeomorphic to a metric space.
Difference between complete metric space and completely metrizable space.
Every discrete space is metrizable by the discrete metric.
The real line with the lower limit topology is not metrizable.
A linear operator on a metrizable vector space is bounded if and only if it is continuous.
The rationals are characterized topologically as the unique countable metrizable space without isolated points.
Locally metrizable space.
A Polish space is a second countable topological space that is metrizable with a complete metric.
Thus every completely metrizable topological space is a Baire space.
See also
Complete metric space Completely uniformizable space Metrizable space.
A space is called Polish if it is metrizable with a separable and complete metric.
Every metrizable space is paracompact, hence completely uniformizable.
The Schwartz space is metrizable and complete.
Every metrizable space is first-countable.
Furthermore, the set of all irrationals is a disconnected metrizable space.
Convergence in distribution is metrizable by the Lévy-Prokhorov metric.
The Nagata-Smirnov metrization theorem in topology characterizes when a topological space is metrizable.
In particular, a manifold is metrizable if and only if it is paracompact.
In metrizable spaces, every open set is an Fσ set.
This implies e . g. that every completely metrizable topological vector space is complete.
If X is a metrizable space, then it is polyadic the converse is also true.
Every locally compact group which is second-countable is metrizable as a topological group i . e.
For instance, all metrizable spaces are collectionwise normal.
Every locally compact, locally connected space, normal Moore space is metrizable.
The long line is locally metrizable but not metrizable ; in a sense it is " too long.
Metrizable spaces are collectionwise normal spaces and are hence, in particular, collectionwise Hausdorff.
For example, a compact Hausdorff space is metrizable if and only if it is second-countable.
This states that every Hausdorff second-countable regular space is metrizable.
Completely metrizable space Uniform space e . g.
A Polish space is a second-countable topological space that is metrizable with a complete metric.
In fact, any metrizable Volterra space is Baire.
If the dual X ′ is separable, the weak topology of the unit ball of X is metrizable.
The weak topology of a Banach space X is metrizable if and only if X is finite-dimensional.
One usually requires M and Z to be locally compact, second countable and metrizable.
Every compact metric space ( or metrizable space ) is separable.
In particular, they are locally compact, locally connected, first countable, locally contractible, and locally metrizable.
Every sequential space ( in particular, every metrizable space ) is countably generated.
Every metacompact, separable, normal Moore space is metrizable.
Also, every uncountable, separable, completely metrizable space contains Cantor spaces as subspaces.
Urysohn 's metrization theorem states that every second-countable, Hausdorff regular space is metrizable.
But in a metrizable space, second-countability coincides with being Lindelöf, so X is second-countable.
Not being Hausdorff, X is not an order topology, nor is it metrizable.
Therefore, if X is a metrizable space with a countable basis, one implication of Bing 's metrization theorem holds.
This definition of weak convergence can be extended for S { \ displaystyle S } any metrizable topological space.
