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Metric spacesDefinition.Let be a set. A functionis called a metric if it satisfies the following threeconditions:1.(positive definitness)2.3.A pair, where is a metric on is called a metric space.Examples.0. Any setwith- discrete metric.1.with2. Any subset of with the same metric.3. Uniform metricLet be any set and letDefineParticular cases:Then we getwith the distanceWe get the space of all bounded real sequences. Notation:Will have many more examples later, as the course proceeds.Definition.Letbe a sequence of elements ofDefinition.Letbe a sequence of elements ofWe say thatWe say that.ifis a Cauchy sequence ifDefinition.Letbe a sequence of elements of We say that is a limit point ofifis infinite.Equivalently:is a limit point ofif there exists a subsequenceDefinition.A metric spaceis called complete if every Cauchy sequence converges to a limit.Already know: with the usual metric is a complete space.Theorem.with the uniform metric is complete.Proof.Letbe a Cauchy sequence inthe sequence of real numbersis a Cauchy sequence (check it!).Since is a complete space, the sequence has a limit. DenoteThenSinceis a Cauchy sequence,Re-write it asThis means thatMetric Spaces Page 1and.

The spaceDefinition.with the metric(will check it later!)Additional structures.1. It is andimensional vector space.2. It has an inner productSo0Let us now prove an importantSchwarz inequality.For all ,Proof.Consideris positive quadratic function of . Let us plug into getNow we can check thatis indeed a metric.Proof.Positive definiteness and symmetry are obvious.Notice that by Schwarz inequalityorTaking the square root, we getwhich implies triangle inequality.Lemma.(Convergence in)A sequence of vectorsis a Cauchy sequence iff all cooordinate sequences are Cauchysequences.A sequence of vectorsconverges to a vector iff all coordinate sequences converge to thecorresponding coordinate ofProof.See the book.Corollary.is complete.Metric Spaces Page 2

Topology of metric spaceDefinition.Letbe a metric space. An open ball of radiusDefinition.Letbe a metric space,centered atis defined asDefine:- the interior of .- the exterior of .- the boundary ofExamples.1. If has discrete metric,2. If is the real line with usual metric,, thenRemarks.2.Definition.is called open ifis called closed ifLemma.is open iffis closed.Proof.is open iffiffLemma.Union of any number of open sets is open.Intersection of finitely many open sets is open.Proof.Letbe any collection of open sets.IfIfis a finite collection of open sets, thenLetiff, sois closed.is open.Then. Sois open.Corollary.Intersection of any number of closed sets is closed.Union of finitely many closed sets is closed.Proof.We just need to use the identitiesExamples.1.is open for allProof.by triangle inequality.2.Proof.are open,is closed.S so it is open.Finally,Definition.The closure of a set is defined asMetric Spaces Page 3, so it is open as a union of open sets.so it is closed.

The closure of a set is defined asTheorem. (Alternative characterization of the closure).iff( is a limit point of ).Proof.Note thatiffIfthensoThusOn the other hand, let. Fix thenTake.SinceYet another characterization of closure.FordefineTheniffRemark.iff is closed.Lemma.is closed. If is closed,then.Proof.Note thatso it is closed as a compliment of an open set.IfthensoRemark.Thus we have another definition of the closed set: it is a set which contains all of its limit points.Lemma.Letbe a complete metric space,.is a complete metric space iff is closed inProof.Assume that is closed in Letbe a Cauchy sequence,Since is complete,But is closed, soOn the other hand, letbe complete, and let be a limit point of so(in ),. Thenis convergent, so it is Cauchy, so it converges in So. Thus contains all ofits limit points, so it is closed.Definition.is called bounded, ifRemark (Hausdorff metric).For a metric spacelet us consider the spacewith the following metric:Check that it is well-defined and a metric!Metric Spaces Page 4of all nonempty closed bounded subset of

Continuous functionsLetandDefinition.Let. LetRemark.Example.be two metric spaces.be a mapping fromif0 does not have to be defined attoWe say thatis a limit of at ,.Characterization of the limit in terms of sequences.iff for every sequenceProof.LetandFixandCombining these two assertions we get.Assume thatwhenIt means thatwe haveThenIn particular, it means thatThusDefinition.is called continuous atifatSequential definition of continuityis continuous at iff for every sequenceProof.The same as for the limit.Topological definition of continuity.is continuous at iffOtherwise,is called discontinuouswe haveExamples.1. Identity function is continuous at every point.2. Every function from a discrete metric space is continuous at every point.3. The following function onis continuous at every irrational point, and discontinuous at every rational point.4.is discontinuous at every point as a function on , but continuous at every point as a functionon5. Let be the usual spacewith the standard metric, and be the same space with theuniform metric. Then the mapis continuous as a functionandcheck it!Definition.is called a continuous function onifis continuous at every point ofTopological characterization of continuous functions.is a continuous function on iff- open, the setMetric Spaces Page 5is open in

is a continuous function on iff- open, the setis open inProof.If is continuous, andthenandsoThus).On the other hand,is an open set, sois an open set which containsit also contains some ball centered atMetric Spaces Page 6Thus

CompactnessIntuitively: topological generalization of finite sets.Definition.A metric spaceis called sequentially compact if every sequenceof elements of has alimit point in . Equivalently: every sequence has a converging sequence.Example:A bounded closed subset of is sequentially compact, by Heine-Borel Theorem.Non-example:If a subset of a metric space is not closed, this subset can not be sequentially compact: just considera sequence converging to a point outside of the subset!Definition.Letbe a metric space. A subsetis called -net ifA metric spaceis called totally bounded iffinite -net.Example:Any bounded subset ofNon-examples.1. Any unbounded set.2. Consider the following subset of :. is bounded, but not totallybounded.Proof.Denote by an element of which is a sequence with in -th position, and in all others.Note thatif. Thus can not have a finite-net!3. Infinite space with discrete topology (but any finite space is totally bounded!)Definition.Open cover of a metric spaceis a collectionof open subsets of , such thatThe spaceis called compact if every open cover contain a finite sub cover, i.e.if we can coverby some collection of open sets, finitely many of them will already cover it!Equivalently:is compact if any collection of closed sets has non-empty intersection if anyfinite sub collection has non-empty intersection. (For the proof, just pass to the complements).Example:Any finite set.Non-examples.1. Any unbounded subset of any metric space.2. Any incomplete space.Turns out, these three definitions are essentially equivalent.Theorem.The following properties of a metric spaceare equivalent:1.is compact.2.is sequentially compact.3.is complete and totally bounded.Proof.Assume thatis not sequentially compact. Letbe a sequence without limit points. Thenall the setsare closed, finitely many of them have non-empty intersection, and-contradiction!A limit point of a Cauchy sequence is its limit (check it!), sois complete if it is sequentiallycompact.Assume now that for somethere is no finite -net. It means that one can inductivelyMetric Spaces Page 7

Assume now that for somethere is no finite -net. It means that one can inductivelyconstruct a sequencesuch that ()ifThis sequence does not have a limitpoint, because for anycontains onlyone member of the sequence - contradiction.(The most interesting part of the proof. It is helpful to compare with the proof of HeineBorel Theorem).Letbe an open cover without finite sub covers. Call a set bad if no finite sub collection ofcovers it. Thus we assumed that itself is bad. Notice another property of bad set: if afinite number of other sets covers a bad set, one of them should be bad.Since there is a finite-net, one can find some bad ball. Because there is a finitenet, one can find some bad ballintersecting the first one. Thus we can inductivelyconstruct a sequence of bad balls, such thatSinceis a Cauchy sequence, so, by completeness of it has a limitfor some, sinceis a cover of Sinceis open,for someNow find a large such that, and. It means thatsois covered by one set from, so it can not be bad - contradiction!Metric Spaces Page 8

Properties of compact sets1. A subset ofis compact iff it is bounded and closed. (Since totally bounded is the same asbounded in ).2. If is compact, andis a continuous map, thenis also compact.Proof.Letbe an open cover ofThenis an open cover of Bycompactness of , it has a finite sub coverThenis a finite open cover of3. Letbe a continuous function. Then is bounded (i.e.is a bounded set).Moreover, it reaches its maximum and minimum on, such that for anywe haveProof.is compact subset of , so it is closed and bounded.Thus, soSimilarly,By the definition of supremum and infimum, for anywe have4. Uniform continuity.Definition.Letandbe two metric spaces.Metric Spaces Page 9is called uniformly continuous ifRemark.It is stronger then usual continuity at every point because here depends only on the andnot on the pointNon-example.is continuous at every point of but not uniformly continuous!Theorem.Every continuous function on a compact set is uniformly continuous.Proof.Let be a continuous but not uniformly continuous function on compact spaceSinceis not uniformly continuous,Sequencehas a subsequence, subsequenceconverging to. Sincealso converges to By continuity of atButso these twosequences can not have the same limit - contradiction!

Connected metric spacesDefinition.A metric spaceis called disconnected if there exist two non empty disjoint open sets: such that.is called connected otherwise.The main property.Ifis a continuous function, thenis connected.Proof.If,thenSinceconnected, one of the setsandis empty. Thus either or is empty.Examples.1. Any discrete compact space with more than one element is disconnected.2.is not connected. ().3. A subset of real line is said to have intermediate point property if.Metric Spaces Page 10isLemma 1.Nonempty subset of the real line has intermediate point property iff it is a point, an interval, aray, or the whole real line.Lemma 2.Subset of the real line is connected iff it has an intermediate point property.Corollary.Nonempty subset of the real line is connected iff it is a point, an interval, a ray, or the wholereal line.Proof of Lemma 1.Clearly all the sets mentioned in the statement satisfy intermediate point property.There are four possibilities: is bounded both above and bellow, is bounded above but notbellow, is bounded bellow but not above, is not bounded above or bellow. I will consideronly the first case, others are done the same way.LetIfthen is just one point.Letand letThen, sinceandSame way,ThusThus, by intermediate valueproperty,We just proved thatThus is an interval (open,closed, semi-open, or semi-closed) with endpoints andProof of Lemma 2.First assume that does not have the intermediate point property, i.e. we can findBut then bothandare not empty (), open, andThus isdisconnected.Assume that has an intermediate point property, and assume thatwhere and are nonempty open sets. Letand let, say,(one of the two numbers has to be larger).Since has an intermediate value property,This means thatLetAssume thatThen(since). Since is open,, so- can not happen because is an upper bound forThusThen(since). Since is open,, so there are no- can not happen becauseThus we arrive to a contradiction,which shows that is connected.Theorem (Intermediate Value Theorem).Letbe a connected metric space, andandThenbe a continuous function. Letfor some