A *homogeneous continuum* is a compact connected metric space X such that for any two points x,y there is a homeomorphism of X taking x to y. This obviously implies that X is locally the same everywhere (*a priori*, it is a stronger condition). There are plenty of examples in books on general topology. My favorite one is a solenoid, which is not a manifold because, for example, it is not locally connected.

**ADDENDUM** The Menger curve C (also known as the Menger sponge, Menger universal curve, and Sierpinski universal curve) is a one-dimensional locally connected continuum. R.D. Anderson proved a characterization which implies that C is n-point homogeneous and that, moreover, up to a homeomorphism, the circle and C are the only one-dimensional homogeneous locally connected continua.

Anderson, R. D. *A characterization of the universal curve and a proof of its homogeneity.* Ann. of Math. (2) 67 1958 313-324 MR

Anderson, R. D. *One-dimensional continuous curves and a homogeneity theorem.*
Ann. of Math. (2) 68 1958 1-16 MR

By the way, I am not a general topologist: all information can be easily found using web searches starting with "homogeneous continuum".

wouldbe called a manifold in the literature. Take a google scholar search of "fractal manifold". Things that look locally like a Sierpinski gasket would be Sierpinski gasket manifolds. FYI it doesn't take much to make the Sierpinski space into a locally homogeneous space I believe removing the three extremal vertices should do the job. $\endgroup$ – Ryan Budney Aug 24 '10 at 17:53