Introduction to Topology

Fundamental problem

A topological space is a set

$X$ together with a collection

$\mathcal{T}$ of subsets of

$X$ called "open" sets such that:

1. the intersection of two open sets is open;
2. the union of any collection of open sets is open; and
3. the empty set
   $\varnothing$ and whole space
   $X$ are open.

Additionally, a subset

$C\subset X$ is called "closed" if its complement

$X\setminus C$ is open.

For example, given a set

$A=\{1,2,3\}$

1. discrete topology:
   $\mathcal{T}=\mathcal{P}(A)$
2. trivial topology:
   $\mathcal{T}=\{\varphi ,A\}$
3. $\mathcal{T}=\{\{1\},\{1,2\},A,\varphi \}$

If

$X$ is a topological space and

$x\in X$ then a set

$N$ is called a neighborhood of

$x$ in

$X$ if there is an open set

$U\subset N$ with

$x\in U$.

If

$X$ is a topological space,

${\left(x_i\right)}_{i=1}^{\mathrm{\infty }}$ is a sequence of points in

$X$, and

$x\in X$, we say that the sequence converges to

$x$, if for every neighborhood

$U$ of

$x$ there exists

$N\in \mathbb{N}$ such that

$x_i\in U$ for all

$i\ge N$.

Continuous

The following definition is a local property.

If

$X$ and

$Y$ are topological spaces and

$f:X\to Y$ is a function, then

$f$ is said to be continuous if

$f^{-1}(U)$ is open for each open set

$U\subset Y$. A map is a continuous function.

The following definition is a global property.

A function

$f:X\to Y$ between topological spaces is called a homeomorphism if

$f^{-1}:Y\to X$ exists (i.e.,

$f$ is one-one and onto) and both

$f$ and

$f^{-1}$ are continuous. The notation

$X\approx Y$ means that

$X$ is homeomorphic to

$Y$.

Theorem. Let

${\mathcal{T}}_1$ and

${\mathcal{T}}_2$ be topologies on the same set

$X$. Show that the identity map of

$X$ is continuous as a map from

$(X,{\mathcal{T}}_1)$ to

$(X,{\mathcal{T}}_2)$ if and only if

${\mathcal{T}}_1$ is finer than

${\mathcal{T}}_2$, and is a homeomorphism if and only if

${\mathcal{T}}_1={\mathcal{T}}_2$.

For example,

• Let
  $(M,d)$ be a metric space, let
  $c$ be a positive real number, and define a new metric
  $d^{\prime }$ on
  $M$ by
  $d^{\prime }(x,y)=c\cdot d(x,y)$. Then,
  $d$ and
  $d^{\prime }$ generate the same topology on
  $M$.
• Let
  $X$ be any set, and let
  $d$ be the discrete metric on
  $X$. Then,
  $d$ generates the discrete topology.

Enugh?

1. Suppose
   $(X,\mathcal{T})$ a topology space where
   $X=\{a,b\}$ and
   $\mathcal{T}=\{\{a\},X,\varphi \}$. Then,
   $\{a\}$ is not closed.
2. Suppose
   $(X,\mathcal{T})$ a topology space where
   $X=\{1,2,3\}$ and
   $\mathcal{T}=\{\{1\},\{1,2\},X,\varphi \}$. Then,
   $x_n=2\to 3$.

Not to structures like a metric, but to conditions describable in terms of the topology itself.

The separation axioms:

$\left({\mathrm{T}}_0\right)$ A topological space

$X$ is called a

${\mathrm{T}}_0$-space if for any two points

$x\ne y$ there is an open set containing one of them but not the other.

$\left({\mathrm{T}}_1\right)$ A topological space

$X$ is called a

${\mathrm{T}}_1$-space if for any two points

$x\ne y$ there is an open set containing

$x$ but not

$y$ and another open set containing

$y$ but not

$x$.

$\left({\mathrm{T}}_2\right)$ A topological space

$X$ is called a

${\mathrm{T}}_2$-space or Hausdorff if for any two points

$x\ne y$ there are disjoint open sets

$U$ and

$V$ with

$x\in U$ and

$y\in V$.

$\left({\mathrm{T}}_3\right)$ A

${\mathrm{T}}_1$-space

$X$ is called a

${\mathrm{T}}_3$-space or regular if for any point

$x$ and closed set

$F$ not containing

$x$ there are disjoint open sets

$U$ and

$V$ with

$x\in U$ and

$F\subset V$.

$\left({\mathrm{T}}_4\right)$ A

${\mathrm{T}}_1$-space

$X$ is called a

${\mathrm{T}}_4$-space or normal if for any two disjoint closed sets

$F$ and

$G$ there are disjoint open sets

$U$ and

$V$ with

$F\subset U$ and

$G\subset V$.

For example,

• Every metric space is Hausdorff: if
  $p_1$ and
  $p_2$ are distinct, let
  $r=d(p_1,p_2)$; then the open balls of radius
  $r/2$ around
  $p_1$ and
  $p_2$ are disjoint by the triangle inequality.

Theorem. Let

$X$ be a Hausdorff space.
(a) Every finite subset of

$X$ is closed.
(b) If a sequence

$\left(p_i\right)$ in

$X$ converges to a limit

$p\in X$, the limit is unique.

Too many?

Let

${\left\{X_{\alpha }\right\}}_{\alpha \in J}$ be an indexed family of topological spaces. Let us take as a basis for a topology on the product space

$$\prod _{\alpha \in J}{X}_{\alpha }$$
the collection of all sets of the form

$$\prod _{\alpha \in J}{U}_{\alpha },$$
where

$U_{\alpha }$ is open in

$X_{\alpha }$, for each

$\alpha \in J$. The topology generated by this basis is called the box topology.

Let

$({\mathbb{R}}^{\omega },\mathcal{T})$ be topology space with box topology. Given any sequence

$\{a_i=(a_{i1},a_{i2},\cdots ):i=1,2,\cdots \}$, exist a neighborhood

$U=(-a_{11},a_{11})\times (-a_{22},a_{22})\times \cdots$ such that

$a_i$ cannot converge to

$a$.

Basis

Let

$X$ be a topological space. A collection

$\mathcal{B}$ of subsets of

$X$ is called a basis for the topology of

$X$ (plural: bases) if the following two conditions hold:
(i) Every element of

$\mathcal{B}$ is an open subset of

$X$.
(ii) Every open subset of

$X$ is the union of some collection of elements of

$\mathcal{B}$.

For example,

• Let
  $M$ be a metric space. Every open ball in
  $M$ is an open subset, and every open subset is a union of open balls. Thus the collection of all open balls in
  $M$ is a basis for the metric topology.
• If
  $X$ is any set with the discrete topology, the collection of all singleton subsets of
  $X$ is a basis for its topology.

Countablility

The countability axioms.

• A topological space is said to be first countable if each point has a countable neighborhood basis.
• Definition. A topological space is said to be second countable if its topology has a countable basis.

For example,

• If
  $X$ is a metric space and
  $p\in X$, the set of balls
  $B_r(p)$ with rational radii is easily seen to be a neighborhood basis at
  $p$, so every metric space is first countable.
• Every Euclidean space is second countable, because as Exercise 2.42(b) shows, it has a countable basis.

The properties of first countable

Lemma (Sequence Lemma). Suppose

$X$ is a first countable space,

$A$ is any subset of

$X$, and

$x$ is any point of

$X$.
(i)

$x\in \overline{A}$ if and only if

$x$ is a limit of a sequence of points in

$A$.
(ii)

$A$ is closed in

$X$ if and only if

$A$ contains every limit of every convergent sequence of points in

$A$.

The properties of second countable

Theorem. Suppose

$X$ is a second countable space.
(i)

$X$ is first countable.
(ii) (separable space)

$X$ contains a countable dense subset.
(iii) (Lindelof space) Every open cover of

$X$ has a countable subcover.

Click here to see exercise in [Rudin]

• 2.22 A metric space is called separable if it contains a countable dense subset. Show that
  $R^k$ is separable.
• 2.23 Prove that every separable metric space has a countable base.
• 2.24 Let
  $X$ be a metric space in which every infinite subset has a limit point. Prove that
  $X$ is separable.

For example, if

${\mathcal{B}}^{\prime }$ is the collection of all half open intervals in the real line,

$$[a,b)=\{x\mid a\le x<b\},$$
the topology generated by

${\mathcal{B}}^{\prime }$ is called the lower limit topology (Sorgenfre line) on

$\mathbb{R}$. Denote it as

${\mathbb{R}}_{\ell }$.

• ${\mathbb{R}}_{\ell }$ is first-countable and separable, but not second-countable.
• Metric space is first countable, but might not be second countble and separable.

Back to Metrization

If

$X$ is a topological space,

$X$ is said to be metrizable if there exists a metric

$d$ on the set

$X$ that induces the topology of

$X$.

${\mathbb{R}}_{\ell }$ is Hausdorff space but not metrizable. Exercise 2.23 in [Rudin] give a intuition that separable metric spaces are second-countable.

When?

Theorem. (The Urysohn Metrization Theorem) Every second countable regular space is metrizable.

Manifold

A topological space

$M$ is said to be locally Euclidean of dimension

$\text{boldsymbol}n$ if every point of

$M$ has a neighborhood in

$M$ that is homeomorphic to an open subset of

${\mathbb{R}}^n$.

An n-dimensional topological manifold is a second countable Hausdorff space that is locally Euclidean of dimension

$n$.

Line with two origins locally Euclidean but non-Hausdorff.

To be continued..

"seperable" is important concept in function space!!

References

1. J. Lee, Introduction to Topological Manifolds
2. G. Bredon, Topology and Geometry
3. J. Munkres, Topology