Under what conditions does a Metrizable space have a Metrizable compactification?

Under what conditions does a metrizable space have a metrizable compactification? SOLUTION. If A is a dense subset of a compact metric space, then A must be second countable because a compact metric space is second countable and a subspace of a second countable space is also second countable.

What is metric space topology?

metric space, in mathematics, especially topology, an abstract set with a distance function, called a metric, that specifies a nonnegative distance between any two of its points in such a way that the following properties hold: (1) the distance from the first point to the second equals zero if and only if the points …

Is compactness a topological?

While metrizability is the analyst’s favourite topological property, compactness is surely the topologist’s favourite topological property. Metric spaces have many nice properties, like being first countable, very separative, and so on, but compact spaces facilitate easy proofs.

What is difference between topology and metric space?

Just in terms of ideas: a metric space has a notion of distance, while a topological space only has a notion of closeness. If we have a notion of distance then we can say when things are close to each other. However, distance is not necessary to determine when things are close to each other.

Why lower limit topology is not metrizable?

The real line with the lower limit topology is not metrizable. The usual distance function is not a metric on this space because the topology it determines is the usual topology, not the lower limit topology. This space is Hausdorff, paracompact and first countable.

What are the types of metric space?


  • 5.1 Complete spaces.
  • 5.2 Bounded and totally bounded spaces.
  • 5.3 Compact spaces.
  • 5.4 Locally compact and proper spaces.
  • 5.5 Connectedness.
  • 5.6 Separable spaces.
  • 5.7 Pointed metric spaces.

What is a Subcover?

subcover (plural subcovers) (topology) A cover which is a subset of another cover. The open intervals cover the real numbers; the open intervals of the form (x, x+1) are a subcover.

Is Cantor set compact?

The Cantor ternary set, and all general Cantor sets, have uncountably many elements, contain no intervals, and are compact, perfect, and nowhere dense.

Why is metric space a topological space?

A subset S of a metric space is open if for every x∈S there exists ε>0 such that the open ball of radius ε about x is a subset of S. One can show that this class of sets is closed under finite intersections and under all unions, and the empty set and the whole space are open. Therefore it’s a topological space.

Is all spaces are metrizable?

One of the first widely recognized metrization theorems was Urysohn’s metrization theorem. This states that every Hausdorff second-countable regular space is metrizable. So, for example, every second-countable manifold is metrizable.