## 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.

**How do you find one point compactification?**

The one-point compactification of a topological space X is a new compact space X*=X∪{∞} obtained by adding a single new point “∞” to the original space and declaring in X* the complements of the original closed compact subspaces to be open.

**Is every Hausdorff space Metrizable?**

Metrization theorems This states that every Hausdorff second-countable regular space is metrizable. So, for example, every second-countable manifold is metrizable. For example, a compact Hausdorff space is metrizable if and only if it is second-countable.

### How do you show a space is metrizable?

A topological space (X,T ) is said to be metrizable if there is a metric d on X that generates T . Due to the fact that very different looking metrics can generate the same topology, we usually talk about metrizable spaces rather than about metric spaces.

**Which topology is metrizable?**

A topology that is “potentially” a metric topology, in the sense that one can define a suitable metric that induces it.

**Why is Q not locally compact?**

It follows that all compact subsets of Q have empty interior (are nowhere dense) so Q can not be locally compact.

## What is compactification string theory?

A compactification of string theory is a solution in which four dimensions of spacetime are macroscopic and the other six form a compact manifold too small to have been observed.

**Are normal spaces metrizable?**

Any metrizable space, i.e., any space realized as the topological space for a metric space, is a perfectly normal space — it is a normal space and every closed subset of it is a G-delta subset (it is a countable intersection of open subsets).

**Are all manifolds metrizable?**

Assuming the usual definition of a topological manifold as a locally Euclidean space which is both Hausdorff and second-countable, it turns out that every manifold M is a metrizable space. This follows from example from Urysohn’s metrization theorem.

### Is every metrizable space normal?

**Is Q locally connected?**

The set of rational numbers Q is neither locally connected nor connected. The same is true for R in the lower limit topology. In fact, a totally disconnected space cannot be locally connected unless it has the discrete topology.

**Which space are locally compact?**

All discrete spaces are locally compact and Hausdorff (they are just the zero-dimensional manifolds). These are compact only if they are finite. All open or closed subsets of a locally compact Hausdorff space are locally compact in the subspace topology.