Definition 1 (Topology).

A topology on a point set $𝒳$ is a collection $𝒰$ of subsets of $𝒳$, which are called open sets, s.t.

1. $𝒳$ is open, the empty set $\mathrm{\varnothing }$ is open.

2. The intersection of any finite collection of open sets is open.

3. The union of any collection (possibly infinite) of open sets is open.

Note: The “point set” may differ from the regular point sets. Maybe a “point” is better to be understand as an “element”. The definition of “open sets” also differs from the regular definition in set theory. Instead, it follows from the three rules listed above.

Definition 2 (Topological Space).

A topological space is a point set $𝒳$ endowed with a topology $𝒰$, denoted as $\mathrm{(}𝒳\mathrm{,}𝒰\mathrm{)}$. Sometimes we omit $𝒰$ and refer to $𝒳$ as a topological space.

Example.

1. Trivial Topology. The point set $𝒳$ together with the empty set forms the trivial topology $\{𝒳,\mathrm{\varnothing }\}$. It is the coarsest topology on $𝒳$.

2. Discrete Topology. The collection of all subsets of $𝒳$ forms the discrete topology $\{u|u\subset 𝒳\}$. It is the finest topology on $𝒳$. Let $𝒳\mathrm{=}\mathrm{\{}\mathrm{1}\mathrm{,}\mathrm{2}\mathrm{,}\mathrm{3}\mathrm{\}}$. Then the discrete topology on $𝒳$ is

   $$\{\mathrm{\varnothing },\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}\}$$

3. Standard Topology on $ℝ$. The unions of open intervals on $ℝ$ forms the standard topology. First, $ℝ$ and $\mathrm{\varnothing }$ are both open intervals. Second, the intersection of a finite collection of unions of open intervals is a union of open intervals, i.e. they are closed to finite intersection. Thrid, the union of a collection of unions of open intervals is a union of open intervals, i.e. they are closed to arbitrary unions.

   

4. A counter example: closed intervals (or unions thereof) do not form a topology (are not open sets) on $ℝ$. Because neither $𝒳$ nor $\mathrm{\varnothing }$ are closed intervals (open sets).

Definition 3 (Neighborhood).

An open neighborhood of a point $x\mathrm{\in }𝒳$ is an open set $u\mathrm{\in }𝒰$, such that $x\mathrm{\in }u$. Similarly, for a subset $𝒜\mathrm{\subset }𝒳$, $u\mathrm{\subset }𝒳$ is an open neighborhood of $𝒜$ if $u$ is open (i.e. $u\mathrm{\in }𝒰$) and $𝒜\mathrm{\subset }u$. A neighborhood of a point (or a subset) is a set containing some open neighborhood of the point (or subset).

Example.

Consider the standard topology on $ℝ$. The open set $\mathrm{(}\mathrm{-}\mathrm{1}\mathrm{,}\mathrm{1}\mathrm{)}$ contains the point $x\mathrm{=}\mathrm{0}$, thus is an open neighborhood around $\mathrm{0}$. The closed interval $\mathrm{[}\mathrm{-}\mathrm{1}\mathrm{,}\mathrm{1}\mathrm{]}$ contains $\mathrm{0}$ and $\mathrm{(}\mathrm{-}\mathrm{1}\mathrm{,}\mathrm{1}\mathrm{)}$, thus is a neighborhood around $\mathrm{0}$. However, since it is not an open set, it is not an open neighborhood.