Compact surfaces, possibly with boundary, are simply closed surfaces with a finite number of holes (open discs that have been removed). Thus, a connected compact surface is classified by the number of boundary components and the genus of the corresponding closed surface – equivalently, by the number of boundary components, the orientability, and Euler characteristic. The genus of a compact surface is defined as the genus of the corresponding closed surface. This classification follows almost immediately from the classification of closed surfaces: removing an open disc from a closed surface yields a compact surface with a circle for boundary component, and removing k open discs yields a compact surface with k disjoint circles for boundary components. The precise locations of the holes are irrelevant, because the homeomorphism group acts k-transitively on any connected manifold of dimension at least 2. Conversely, the boundary of a compact surface is a closed 1-manifold, and is therefore the disjoint union of a finite number of circles; filling these circles with disks (formally, taking the cone) yields a closed surface. The unique compact orientable surface of genus g and with k boundary components is often denoted for example in the study of the mapping class group. Non-compact surfaces Non-compact surfaces are more difficult to classify. As a simple example, a non-compact surface can be obtained by puncturing (removing a finite set of points from) a closed manifold. On the other hand, any open subset of a compact surface is itself a non-compact surface; consider, for example, the complement of a Cantor set in the sphere, otherwise known as the Cantor tree surface. However, not every non-compact surface is a subset of a compact surface; two canonical counterexamples are the Jacob's ladder and the Loch Ness monster, which are non-compact surfaces with infinite genus. A non-compact surface M has a non-empty space of ends E(M), which informally speaking describes the ways that the surface "goes off to infinity". The space E(M) is always topologically equivalent to a closed subspace of the Cantor set. M may have a finite or countably infinite number Nh of handles, as well as a finite or countably infinite number Np of projective planes. If both Nh and Np are finite, then these two numbers, and the topological type of space of ends, classify the surface M up to topological equivalence. If either or both of Nh and Np is infinite, then