**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.

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**.

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.

**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.

**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.

**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.

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**.

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).

**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.

**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.

In topology and related branches of mathematics, a normal space is **a topological space X that satisfies Axiom T _{4}: every two disjoint closed sets of X have disjoint open neighborhoods**. A normal Hausdorff space is also called a T

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.

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.

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