characterized the Sierpiński carpet as follows: any curve that is locally connected and has no 'local cut-points' is homeomorphic to the Sierpinski carpet. Here a local cut-point is a point for which some connected neighborhood of has the property that is not connected. So, for example, any point of the circle is a local cut point. In the same paper Whyburn gave another characterization of the Sierpiński carpet. Recall that a continuum is a nonempty connected compact metric space. Suppose is a continuum embedded in the plane. Suppose its complement in the plane has countably many connected components and suppose: the diameter of goes to zero as ; the boundary of and the boundary of are disjoint if ; the boundary of is a simple closed curve for each ; the union of the boundaries of the sets is dense in . Then is homeomorphic to the Sierpiński carpet. Brownian motion on the Sierpiński carpet The topic of Brownian motion on the Sierpiński carpet has attracted interest in recent years. Martin Barlow and Richard Bass have shown that a random walk on the Sierpiński carpet diffuses at a slower rate than an unrestricted random walk in the plane. The latter reaches a mean distance proportional to after steps, but the random walk on the discrete Sierpiński carpet reaches only a mean distance proportional to for some . They also showed that this random walk satisfies stronger large deviation inequalities (so called "sub-Gaussian inequalities") and that it satisfies the elliptic Harnack inequality without satisfying the parabolic one. The existence of such an example was an open problem for many years. Wallis sieve A variation of the Sierpiński carpet, called the Wallis sieve, starts in the same way, by subdividing the unit square into nine smaller squares and removing the middle of them. At the next level of subdivision, it subdivides each of the squares into 25 smaller squares and removes the middle one, and it continues at the th step by subdividing each square into (the odd squares) smaller squares and removing the middle one. By the Wallis product, the area of the resulting set is , unlike the standard Sierpiński carpet which has zero limiting area. Although the Wallis sieve has positive Lebesgue measure, no subset that is a Cartesian product of two sets of real numbers has this property, so its Jordan measure is zero. Applications Mobile phone and