2.1 Basic concepts of general topology

Definition 2.1.1   Let $X$ be an arbitrary set. Let a collection $\tau$ of subsets $U$ of $X$ be fixed such that

1. $\emptyset, X\in \tau$ ,
2. The union of any collection of sets in $\tau$ is a set in $\tau$ ,
3. The intersection of any finite collection of sets in $\tau$ is a set in $\tau$ .

Then $\tau$ is called a topology in $X$ . The set $X$ together with a fixed topology $\tau$ is called a topological space and is denoted $(X,\tau)$ . The sets $U\in \tau$ are called open.

Example 2.1.2   1. $X$ is the real line ${\mathbb{R}}$ . The topology $\tau$ is given by the empty set $\emptyset$ , all kind of intervals and their unions: $\tau:=\{\emptyset, \bigcup\limits_\alpha (a_\alpha,b_\alpha)\}$ .

2. $X={\mathbb{R}}^2$ . We say that a subset $U\subseteq X$ is right if for any point $x\in U$ the set $U$ contains the inner part of a small disk with the center in $x$ . more formally, this can be determined as follows: there is $r>0$ such that $\{y : \, \vert x-y\vert<r\}\subseteq U$ . The set of all right sets is a topology in $X$ . Let us note that in this topology the open sets are precisely the right sets.

3. Let $X$ be an arbitrary set, $\tau_0:=\{\emptyset, X\}$ . Then $(X,\tau_0)$ is a topological space.

4. Let $X$ be an arbitrary set, $\tau_1$ be a collection of all subsets of $X$ . Then $(X,\tau_1)$ is a topological space.

Exercise 2.1.3   Check that the topologies introduced in the above example are indeed topologies.

Definition 2.1.4   The topology $\tau_1$ is called maximal or discrete, and $\tau_0$ is called minimal or trivial.

Let us point out that a given set can be provided with different topologies, for example trivial and discrete.

Proposition 2.1.5   Different topologies on a given set constitute a partially ordered set with the minimal element $\tau_0$ and the maximal element $\tau_1$ , here $\tau\prec \tau'$ if $U\in \tau$ implies that $U\in \tau'$ .

Definition 2.1.6   Let $(X,\tau)$ be a topological space. A subset $V\subseteq X$ is said to be closed in the topology $\tau$ iff its complement $X\setminus V$ is open in $\tau$ .

Exercise 2.1.7   What are the closed subsets in each of the topologies 1-4 in Example ?

Exercise 2.1.8   Are there sets in a topological space that are both open and closed?

Definition 2.1.9   A neighborhood of a point $x$ in a topological space $(X,\tau)$ is any subset $U(x)\subseteq X$ such that

1. $x\in U(x)$ ,
2. there exists $U\in \tau$ such that $x\in U \subseteq U(x)$ .

Definition 2.1.10   A topological space $(X,\tau)$ is called Hausdorff if for any points $x,y\in X$ , $x\ne y$ , there exist neighborhoods $U(x),U(y)$ of these points such that $U(x)\cap U(y)=\emptyset$ .

Definition 2.1.11   Let $(X,\tau)$ , $(Y,\sigma)$ be topological spaces with topologies $\tau,\sigma$ , correspondingly, $f:X\to Y$ be a transformation of sets. It is said that $f$ is a continuous transformation of topological spaces if for any open set $V$ in $(Y,\sigma)$ its preimage $f^{-1}(V)$ is an open set in $(X,\tau)$ .

Proposition 2.1.12   Let $(X,\tau)$ , $(Y,\sigma)$ , $(Z,\omega)$ be topological spaces, $f:X\to Y$ , $g:Y\to Z$ be transformations. If both $f$ and $g$ are continuous, then their composition $g\circ f$ , which is defined by $(g\circ f)(x)=g(f(x))$ , is continuous.

Exercise 2.1.13   Prove Proposition .

Definition 2.1.14   Topological spaces $(X,\tau)$ and $(Y,\tau)$ are called homeomorphic if there exists a map $f:X\to Y$ such that

1. $f:X\to Y$ is bijective,
2. $f$ is continuous,
3. $f^{-1}$ is continuous.

In this case the transformation $f$ is called a homeomorphism between $(X,\tau)$ and $(Y,\sigma)$ .

Definition 2.1.15   Let $(X,\tau)$ , $(Y,\sigma)$ be topological spaces. A transformation $f:X\to Y$ is called open (resp., closed) if the image of any open (resp., closed) set in $(X,\tau)$ is an open (resp., closed) set in $(Y,\sigma)$ .

Exercise 2.1.16   Prove that a transformation $f:X\to Y$ is a homeomorphism if and only if the transformation $f^{-1}: Y\to X$ exists and both transformations $f$ and $f^{-1}$ are open and closed.

Definition 2.1.17   Properties of topological spaces which are not changed under homeomorphisms are called topological properties.

Definition 2.1.18   Let $(X,\tau)$ be a topological space, $Y\subseteq X$ be a subset of $X$ . Let $\tau_Y$ be the collection of subsets of $Y$ which consists of those and only those $V\subseteq Y$ such that $V=U\cap Y$ for some $U\in \tau$ . This $\tau_Y$ is called an induced topology from $(X,\tau)$ . Such topological space $(Y, \tau_Y)$ is called a topological subspace of $(X,\tau)$ .

Exercise 2.1.19   Prove that $\tau_Y$ is a topology in $Y$ .

Definition 2.1.20   Let $X$ be a set, $\delta$ be a certain collection of subsets $A\subseteq X$ . Then $\delta$ is called a covering of a subset $Y\subseteq X$ if $\bigcup\limits_{A\in\delta} A\supseteq Y$ . It is said that a covering $\delta'$ is a sub-covering of $\delta$ if any element of $\delta'$ is contained in a certain element of $\delta$ .

Definition 2.1.21   A covering $\delta$ is called open in the topology $\tau$ in $X$ if all the sets $A\in \tau$ .

Definition 2.1.22   Let $(X,\tau)$ be a Hausdorff topological space. A subspace $(Y, \tau_Y)$ of $(X,\tau)$ is called compact if for any open covering $\delta$ of $Y$ there exists a finite (consisting of a finite number of sets $A$ ) open sub-covering $\delta'$ of $\delta$ , which is a covering of $Y$ .

Definition 2.1.23   A topological space $(X,\tau)$ is called disconnected if there are $X_1,X_2\in \tau$ such that $X_1\ne \emptyset$ , $X_2\ne \emptyset$ , $X_1\cap X_2=\emptyset$ , $X_1\cup X_2=X$ . Otherwise $(X,\tau)$ is called connected.

Control question 2.1   Provide an example of a set $X$ with two different topologies $\tau_1$ , $\tau_2$ such that $(X,\tau_1)$ is a connected topological space and $(X,\tau_2)$ is a disconnected topological space.

Control question 2.2   Provide a definition of topological space based on the usage of the concept of closed sets.

Please, see the answer form for Control question  to answer these questions.

Definition 2.1.24   It is said that sets $X_1$ , $X_2$ constitute a disjoint union if $X_1\cap X_2=\emptyset$ .

For more detailed and deep introduction to topology you may see for example
http://en.wikipedia.org/wiki/Topological_space
or read the books

[1]

Bredon, Glen E., Topology and Geometry (Graduate Texts in Mathematics), Springer; 1 edition (October 17, 1997). ISBN 0387979263.

[2]

Bourbaki, Nicolas; Elements of Mathematics: General Topology, Addison-Wesley (1966).

[3]

Dubrovin Boris A., Novikov Sergei P., Fomenko Anatolii T.; Modern Geometry, Science (1979) [in Russian].

[4]

Fulton, William, Algebraic Topology, (Graduate Texts in Mathematics), Springer; 1 edition (September 5, 1997). ISBN 0387943277.

[5]

Lipschutz, Seymour; Schaum's Outline of General Topology, McGraw-Hill; 1st edition (June 1, 1968). ISBN 0070379882.

[6]

Mischenko Alexander S., Fomenko Anatolii T., The course of differential geometry and topology, 3 edition, Factorial (2000) [in Russian].

To go back to the main course, please go to Section .