Using the topologist sine curve we present a new functorial construction of cone-like spaces, starting in the category of all path-connected topological spaces with a base point and continuous maps, and ending in the subcategory of all simply connected spaces. If one starts by a noncontractible n-dimensional Peano continuum for any n>0, then our construction yields a simply connected noncontractible (n + 1)-dimensional cell-like Peano continuum. In particular, starting with the circle $\mathbb{S}^1$, one gets a noncontractible simply connected cell-like 2-dimensional Peano continuum.
It is well known that all cell-like polyhedra are contractible. Griffiths [5] constructed a 2-dimensional nonsimply connected cell-like Peano continuum: Let H 1 be the 1-dimensional Hawaiian earrings with the base point θ at which H 1 is not locally simply connected. Let Y = C(H 1 ) be the cone over H 1 . Then H 1 can be considered as the base of the cone C(H 1 ) and θ as its base point. The Griffiths space is then defined as the bouquet of two copies of Y with respect to the point θ.
A generalization of the Griffiths example is analogous -instead of the 1dimensional one considers the 2-dimensional Hawaiian earrings [3], i.e. the subspace H 2 of the 3-dimensional Euclidean space,
It is easy to see that this generalization of the Griffiths example is a 3-dimensional noncontractible simply connected cell-like Peano continuum.
The purpose of the present paper is to construct a functor SC(-, -) from the category of all path connected spaces with a base point and continuous mappings, to the subcategory of all simply connected spaces with a base point. The following are our main results: Theorem 1.1. For every path connected space Z with z 0 ∈ Z, the space SC(Z, z 0 ) is simply-connected. Theorem 1.2. For every noncontractible space Z with z 0 ∈ Z, the space SC(Z, z 0 ) is noncontractible.
If Z is a Peano continuum, then SC(Z, z 0 ) is also a Peano continuum. If Z is an n-dimensional metrizable space for n > 0, then the space SC(Z, z 0 ) is (n + 1)-dimensional. If Z is a compact, then SC(Z, z 0 ) is a compact space with trivial shape.
In particular, when Z is the circle S 1 , we get the following:
Corollary 1.3. For any point z 0 of the circle S 1 , the space SC(S 1 , z 0 ) is a noncontractible simply connected cell-like 2-dimensional Peano continuum.
As a general reference for algebraic topology we refer the reader to [10].
For any two points A and B in the plane R 2 , [A, B] denotes the linear segment connecting these points. Let Z be any space with a base point z 0 . Then the base set of SC(Z, z 0 ) is the quotient set of T × Z ∪ I 2 obtained by the identification of the points (s, z 0 ) ∈ T × Z with s ∈ T ⊂ I 2 and by the identification of each set {s} × Z with the one-point set {s} if s ∈ L. There is a natural projection p : SC(Z, z 0 ) → I 2 . To p there corresponds a pair of functions p 1 and p 2 such that p(z) = (p 1 (z); p 2 (z)). For a = (x; y) ∈ T with x > 0, the set p -1 (a) is denoted by Z a , which is homeomorphic to Z, and for y ∈ I the set p -1 2 ({y}) is denoted by M y . Let O ε (a) = p -1 (U ε (a)), where U ε (a) is the open ε-ball with the center at a ∈ I × I with respect to the standard metric.
The topology of SC(Z, z 0 ) coincides with the quotient topology at each point outside L. A basic neighborhood of a point a = (0; y) ∈ L is of the form O ε (a). Therefore, the topology of SC(Z, z 0 ) is the quotient topology when Z is compact.
Obviously, SC(-, -) is a functor from the category of topological spaces with a base point to itself. The space SC(Z, z 0 ) is path-connected, pathconnected and locally connected, finite-dimensional, metrizable or compact if Z is path-connected, path-connected and locally connected, metrizable or compact, respectively. In particular, SC(Z, z 0 ) is a Peano continuum if Z is a Peano continuum.
If Z is compact, SC(Z) is a quotient space of T × Z ∪ I 2 and hence SC(Z) is also compact. Next we show that the shape type of SC(Z) is that of the one-point space, when Z is compact. To see this let U be an open cover of SC(Z). By the compactness of I we have ε > 0 such that p -1 ([0, ε) × [a, b]) is contained in an element of U for every 0 ≤ a < b ≤ 1 with ba < ε. By the compactness of Z we also have a cover
and each [P i , P i+1 ] × O j is contained in an element of U . Hence we have a refinement of U whose nerve is contractible. This yields the conclusion.
Let Z be an n-dimensional metrizable space for n > 0. Then, since [4,Theorems 4.1.3 and 4.1.9] and [9, p.221]. Hence SC(Z) is a cell-like, (n + 1)-dimensional compact metrizable space, if Z is a n-dimensional compact metrizable space [7,8].
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