| The aim of this paper is to present some elementary results from infinite-dimensional topology in Hilbert spae . We present infinite-dimensional topologyin l2 . The main results are by Anderson that are l2 is homeomorphic to R∞, and T×Q is homeomorphic to Q ,where Q is the Hilbert cube ,T is the subspase ([0,1]×{0})×({1/2}×[0,1]) of [0,1]×[0,1], which looks like the letterl2 and R∞are not homeomorphism ,but it is true for the infinite-dimensional space R∞? Frech arised the question that whether l2 is homeomorphic to Rn in 1923 , The topology in R∞is the coordinatewise convergence , it doesn't the same as l2,s topology .For example , en = ((?)1,0,…) convergencesto 0 in l2 ,but it doesn't convergence in R∞. Untill 1966 , Anderson showed l2 and R∞are homeomotphism , T×Q is homeophic to Q .R∞is not normable ,but l2 is a Hilbert space R∞and l2 are both topo-logical complete and local convex .In order to present l2 and R∞are homeomotphism ,we introduce Hilbert cube , which plays an important role in the proof of l2 is homeomotphic R∞In the theory of infinite dimensional topology, Hilbert cube is both strongly infinite dimensional and weakly infinite dimensional, but it is neither countableinfinite dimensional nor hereditarily infinite dimensional .Hilbert cube is Absolute Retracts and is homogeneous .For the aim of this paper ,we introduce three methods to construct new homeomorphism from old ones ,namely, (1) the Inductive Convergence Criterion;(2) Bing's Shrinking Criterion ; (3) by means of isotopies .We usually use homeomorphism extension to construct some new homeophism .The homeomorphismextension for some subsets of Q is important to present the results of this paper ,that are Z-sets;Definition 1 ( Z-sets) Let X be a seperable metric space ,A (?) X is closed , if for allε> 0 , every f∈C(Q, X) , there is g∈C(Q, X) such that(1)d(f,g)<ε,(2) g((Q)∩A = (?).then A is called Z-set, denoted by A∈Z(X). Zσis a countable union of Z-sets .Now we can define absorber ,which is necessary in the proof of l2 and R∞are homeomotphism .Definition 2 ( Absorbers) A∈Zσ(Q), A is called an absorber provided that for all L, K∈Z(Q),everyε> 0 , there is a homeophism h∈H(Q), such that,(1) d(h, 1Q) <ε,(2) h|K= 1,(3) h(L\K) (?)A.We have the following result, which is useful in this paper;Corollary 1 A is an absorber , B∈Zσ(Q) ,then there is h∈H(Q) ,such taht h(A) = A∪B .For the Z-sets in Q ,we have the extension theorem;Theorem 1 Let E,F∈Z(Q),and let f : E→F be a homeophism such that d(f,1E) <ε,then f can be extended to a homeomorphism f : Q→Q , such that d(f, 1) <ε.Let K = {x∈R∞:sum from n=1 to∞xn2≤1},it is the compactification of l2 , if K ishomeophic to Q , and K\l2 is an absorber in K, then we yield l2≈R∞. This is the key of this paper.Theorem 2 (Anderson) l2≈R∞.At last, we present the definition of inverse limits, an application of Brown's Appromiate Theorem, then yield T×Q≈Q .Definition 3 (Inverse Limits) Let Xn be a sequence of space , fn : Xn+1→Xn is continuous functions , this sequence of pairs (Xn,fn)n≥1 of Xn is called an inverse sequence. The inverse limit of the inverse sequence (Xn, fn)nis defined as the following subspase of Xn;denoted by X∞= lim(Xn,fn)n , the restriction to lim(Xn, fn)nof the projectiononto the n-factor of the product multiply fromn=1 to∞Xn shall be denoted by f(∞,n.定ç†3 (Brown's Appromiate Theorem ) Let (Xn, fn)n be an inverse sequence consisting of compact space with inverse limit X∞.If each fn is a near homeophism , then so is f∞,n, X∞is homeophic to X1 .
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