Topology

A topological space is a set with the least structure necessary to define the concepts of nearness and continuity; you have met examples in real and complex analysis and perhaps also as a metric space (a set with the least structure necessary to support the concept of distance). General topology is concerned with the study of topological spaces and maps among them while algebraic topology is concerned with the casting of topological problems into easier algebraic form using functors. Definitions
A metric space (X,d) is a nonempty set X and a map $d : X \times X \to \mathbb{R} ,$ called a metric or Hausdorff distance function, satisfying the natural requirements of a distance function:

1.

$d(x,y) \ = \ d(y,x) \quad \forall\, x,y \in X$    (Symmetry)

2.

$d(x,y) \ \geq \ 0 \mbox{ and } d(x,y) \ = \ 0 \ \Longleftrightarrow \ x = y$    (Positive definiteness)

3.

$d(x,z) \leq d(x,y) + d(y,z)$    (Triangle Inequality)

The standard metric on a normed vector space is simply the norm of the difference between two points. In geometry, Euclidean n-space, $\mathbb{E} ^n,$ is the metric space of points in $\mathbb{R} ^n$ with distance function

d(p,q)=||q-p||.

A topological space $(X, {\cal T})$ is a set X together with a collection ${\cal T}$ of subsets, so ${\cal T}\subseteq P(X)$, satisfying:

1.

$\emptyset, X \in {\cal T}$

2.

${\cal T}$ is closed under finite intersections

3.

${\cal T}$ is closed under arbitrary unions.

We call ${\cal T}$ the topology of the space $(X, {\cal T})$ or a topology on the set X. Elements of ${\cal T}$ are called open sets  in the topological space $(X, {\cal T})$ or they are called ${\cal T}$-open sets of X. A base for a topology ${\cal T}$ on X is a collection ${\cal B}\subseteq {\cal T}$ of open sets of X such that every member of ${\cal T}$ is expressible as a union of members of ${\cal B}.$ Every metric space (X,d) has a topology ${\cal T}_d$ determined by d. A subset A of X is d-open in (X,d) if it contains an open ball around each of its points, and we define ${\cal T}_d$ to be the set of d-open subsets. So a base for a metric topology is the collection of all open balls. Let $(X, {\cal T})$ be a topological space. A set A is closed  in $(X, {\cal T})$ if $X \setminus A$ is open in $(X, {\cal T})$ (that is, closed if it is the complement of an open set). Sometimes $X \setminus A$ is denoted X-A. A point $x \in X$ is a limit point  of $A \subseteq X$ in $(X, {\cal T})$ if every neighborhood of x meets $A \setminus \{x\}$ non-emptily. A limit point of A need not be in A, but it turns out that A is closed in $(X, {\cal T})$ if and only if A contains all of its limit points. The closure  $\bar{A}$ of a set A in a topological space is the union of A with all of its limit points; that is, the smallest closed set containing A. The interior  , ${\rm int \ }A$ (also denoted $(A)^\circ$, when convenient) of A is the largest open set contained in A. A is dense in  $(X, {\cal T})$ if $\bar{A} = X$. The boundary  or frontier  of a set A is $\partial A = \bar{A} \cap \overline{X\setminus A}$. A map between topological spaces $f : (X,{\cal T}) \to (Y, {\cal T}')$ is called continuous  if

$$\forall\, B \in {\cal T}', \ f^\leftarrow B \in {\cal T}.$$

A continuous map f is called
open if $U \in {\cal T}\ \Rightarrow \ fU \in{\cal T}'$
and is called
closed if $U \in {\cal T}\ \Rightarrow \ f(X\setminus U)$ is closed in Y.
Open, closed, and continuous are independent properties. A map $f:X \to Y$ is called a homeomorphism if

$$f \hbox{ is continuous,}\quad f^{-1} \hbox{ exists, and} \quad f^{-1} \hbox{ is continuous;}$$

that is, if f is bijective and bicontinuous; then we say that X and Y are homeomorphic and write $X\cong Y$. For proofs in topology we usually juggle the properties of open and closed sets. We can get open sets in relatively few ways:

1.

directly from ${\cal T}$;

2.

as complements of closed sets;

3.

as inverse images of open sets by continuous maps.