# Are Metric Spaces First Countable?

Every metric space is first-countable. For x ∈ X, consider the neighborhood basis Bx = {Br(x) | r > 0,r ∈ Q} consisting of open balls around x of rational radius.

### Is usual topology connected?

A subset of a topological space is called connected if it is connected in the subspace topology. R with its usual topology is not connected since the sets and are both open in the subspace topology. R with its usual topology is connected.

### Is every metric space Lindelof?

Properties of Lindelöf spaces

Every second-countable space is Lindelöf, but not conversely. For example, there are many compact spaces that are not second countable. A metric space is Lindelöf if and only if it is separable, and if and only if it is second-countable. Every regular Lindelöf space is normal.

### Is vector space connected?

A space is connected if it cannot. 2,3) tu1+(1−t)u3=(1−t,0,0) which is a vector in V. R3∖V∪({(1−t,0,0)|0≤t≤1}) is a union of disjoint sets.

### Are metric spaces first countable?

Every metric space is first-countable. For x ∈ X, consider the neighborhood basis Bx = {Br(x) | r > 0,r ∈ Q} consisting of open balls around x of rational radius.

### Is the closure of a connected set connected?

The closure of a connected set is connected. (Proof: Assume that the closure of S is the union of two disjoint nonempty closed sets A and B. If S is connected then one of the sets A∩S and B ∩S must be S, say the former.

### Is the empty space connected?

With the common naive definitions that “a space is connected if it cannot be partitioned into two disjoint nonempty open subsets” and “a space is path-connected if any two points in it can be joined by a path,” the empty space is trivially both connected and path-connected.

### How do you find simply connected?

A region D is said to be simply connected if any simple closed curve which lies entirely in D can be pulled to a single point in D (a curve is called simple if it has no self intersections).

### Is every indiscrete space connected?

Every indiscrete space is connected. Let X be an indiscrete space, then X is the only non-empty open set, so we cannot find the disconnection of X. Hence X is connected. A subspace Y of a topological space is said to be a connected subspace if Y is connected as a topological space in its own right.

### Is connected set open?

A connected set is a set that cannot be partitioned into two nonempty subsets which are open in the relative topology induced on the set. Equivalently, it is a set which cannot be partitioned into two nonempty subsets such that each subset has no points in common with the set closure of the other.

### What does normal space refer to?

In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T4: every two disjoint closed sets of X have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space.

### Is trivial topology hausdorff?

In particular, it is not a Hausdorff space. Not being Hausdorff, X is not an order topology, nor is it metrizable. X is, however, regular, completely regular, normal, and completely normal; all in a rather vacuous way though, since the only closed sets are ∅ and X.

### Is every locally path connected space is path connected?

Proof. A path connected component is always connected (this lemma), and in a locally path-connected space is it also open (lemma ). This means that every path-connected component is also connected. Conversely, it is now sufficient to see that every connected component is path-connected.

### What is discrete and indiscrete?

Discrete and the Indiscrete Topology

It is defined as follow: Let X be a non-empty set and let T be the collection of all subset of X i.e T is the power set of X. Then we say that T is the discrete topology on X and the pair (X,T) is called a discrete topological space.

Dated : 24-Jun-2022

Category : Education