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## 2.1 Topological spaces

A topological space is a set X together with a collection of subsets OS the members of which are called open, with the property that (i) the union of an arbitrary collection of open sets is open, and (ii) the intersection of a finite collection of open sets is open. (In particular X is open, as is the empty set.)

Examples (i) Metric spaces: there is a realvalued distance function on X × X satisfying the triangle inequality; an open set is a set containing balls with sufficiently small radius around each of its points.
(ii) Ordered spaces: there is a total order on X; an open set is a set containing for each point y an entire orderinterval (x,z), where x < y < z.
(iii) Discrete topology: every subset of X is open.
(iv) Indiscrete topology: the only open subsets of X are the empty set and X itself.

A map or continuous function from a topological space (X,OX) to a topological space (Y,OY) is a function from X to Y such that the preimage of any member of OY is a member of OX.

A homeomorphism is a bijective map of which the inverse is a map, too.

A neighbourhood of a set is an open set containing it.

A closed set is the complement of an open set.

A basis for the open sets is a collection of open sets such that each open set is a union of some subcollection. A subbasis for the open sets is a collection of open sets such that one obtains a basis by taking finite intersections. In order to check that a given function is continuous, it suffices to check that the inverse images of the members of a subbasis for the open sets are open again.

The interior of a set A is the largest open set contained in A. The closure of A is the smallest closed set containing A. The boundary of A is the intersection of its closure and the closure of its complement.

Example Let (X,OX) be the closed unit interval [-1,1] with usual topology and let (Y,OY) be the space obtained from it by identifying the points 0 and 1. Let p be the quotient map. Then the restriction p' of p to the half-open interval [-1,1) is a bijection and is continuous, but its inverse is not continuous, so p' is not a homeomorphism.

Exercise (i) Let A be a subset of a topological space X. How many different subsets can I make starting from A and repeatedly using the closure and interior operations? Answer.
(ii) Let A be a subset of a topological space X. How many different subsets can I make starting from A and repeatedly using the boundary and union operations? Answer.

## 2.2 Operations on topological spaces

### Topologies defined by functions

Let A be a set, and F a collection of functions from A to topological spaces. The topology on A defined by F is the weakest topology (i.e., the smallest collection OA) for which all these functions become continuous.

Similarly, let B be a set, and F a collection of functions from topological spaces to B. The topology on B defined by F is the strongest topology (i.e., the largest collection OB) for which all these functions become continuous.

### Subspaces

Given a topological space (X,OX) and a subset A, we may (and will) consider it a topological space in its own right (a subspace) by giving it the topology defined by the inclusion map.

### Product spaces

Given an arbitrary collection of topological spaces (Xi,OXi), their (Cartesian) product is the topological space with as point set the Cartesian product of the sets Xi, and topology defined by the projection maps.

### Quotient spaces

Given a topological space (X,OX) and a function f from X to a set B, we call the topology on B determined by f the quotient topology, and f the corresponding quotient map. Frequently B will be the set of equivalence classes in X of some equivalence relation R. In this case the quotient space is denoted X/R.

## 2.3 Separation

A topological space is called Hausdorff (or (T2)) when any two points have disjoint neighbourhoods. (A weaker requirement called (T1) is that every singleton is a closed subset.) A topological space is called normal when any two disjoint closed sets have disjoint neighbourhoods. A topological space is called metric when there is a distance function determining the topology (i.e., open balls for the metric are open sets, and conversely, if a point x lies in an open set U then for some positive e the ball with radius e around x is contained in U.

A metric space is normal since {x|d(x,A) < d(x,B)} and {x|d(x,A) > d(x,B)} are disjoint neighbourhoods of the disjoint closed sets A and B.

A Hausdorff space X is normal if and only if for each pair of disjoint closed sets A and B there exists a map f from X to the unit interval I that is identically 0 on A and identically 1 on B. (Urysohn)

## 2.4 Covering

A topological space is called compact when every open cover (i.e., covering with open sets) has a finite subcover.

A compact subset of a Hausdorff space is closed.

A closed subset of a compact space is compact.

The continuous image of a compact space is compact again.

A topological space is called paracompact when every open cover has a locally finite refinement.

## 2.5 Convergence

A filterbase is a collection of nonempty sets such that the intersection of any two contains a third. We say that a filterbase converges to a point if each neighbourhood of the point contains an element of the filterbase. We say that a filterbase accumulates at a point if each neighbourhood of the point meets an element of the filterbase.

A space is Hausdorff iff each convergent filterbase converges to exactly one point.

A space is compact iff each filterbase has an accumulation point.

A space is compact iff each maximal filterbase converges.

Exercise Prove: the cartesian product of compact spaces is compact. (Tychonoff)

## 2.6 Connectedness

A topological space is called connected when the only subsets that are both open and closed are the empty set and the entire space. A (connected) component of a topological space is a maximal connected subset.

The continuous image of a connected space is connected again.

### Path-connectedness

In particular, an image of the closed unit interval [0,1] (sometimes called an arc or a path) is connected. A topological space is called path-connected or arcwise connected when any two of its points can be joined by an arc. (Note: below we shall use the word path for the mapping from [0,1] into some space, rather than for its image.)

### Quasi-components

The intersection of all open-and-closed subsets containing a given point x is called the quasi-component of x. The partition of a topological space into quasi-components is coarser than the partition into components.

Exercise Prove: The closure of a connected set in an arbitrary topological space is again connected. Components and quasicomponents of a topological space are closed.

Exercise Prove: Let X be connected, and A a connected subset, and Q either open-and-closed, or a component, or a quasicomponent in X\A. Then X\Q is connected. Answer

Exercise (i) Construct a topological space that is connected but not path-connected. (ii) Construct a topological space with a quasicomponent that is not connected. (iii) Show that in a compact space components and quasicomponents coincide.

Warning There exist strange objects, like connected Hausdorff spaces that become totally disconnected (all components are singletons) when one removes a well-chosen point, or connected metric spaces without nontrivial open connected subspaces. Also, there need not be any non-constant maps from [0,1] to a connected topological space (indeed, there exist countable connected Hausdorff spaces), and in such a case all path-components are singletons.

Problem Show that there exists a connected subset of the Euclidean plane such that all its path-components are singletons.

## 2.7 Local properties

A topological space is said to be locally P for some property P when for each point x and neighbourhood U of x there is a set A contained in U and containing a neighbourhood of x that has property P.

For example, the Euclidean plane is locally compact but not compact.

Exercise (i) Give an example of a space that is connected but not locally connected. (ii) Show that if X is connected and A a compact nonempty proper subspace, then each component of A meets the boundary of A.

## 2.8 Function spaces

Let C(X,Y) be the set of all maps from X into Y. This set has several natural topologies. First of all, there is the product topology, where this is regarded as a subspace of the product of |X| copies of Y. Here a basic open set constrains the values at finitely many points to an open set. Much finer is the compact-open topology (also called c-topology), where a subbasic open set contains the functions mapping a given compact subset of X into a given open subset of Y. Below we shall assume that C(X,Y) has the compact-open topology.

Let Y be locally compact and X,Z Hausdorff. Then the composition map from C(X,Y) × C(Y,Z) to C(X,Z) is continuous. In particular, under these assumptions, the evaluation map from C(Y,Z) × Y to Z is continuous.

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