A closed set in topology is characterized by the property that it contains all of its limit points, meaning that every point that is arbitrarily close to a point in the set is also in the set.

A topological space is a set equipped with a topology, which is a collection of open sets that satisfy certain axioms. The set of real numbers with the usual topology is a topological space because the open sets are defined in a way that satisfies these axioms.

The topological dimension of a space is a topological invariant that characterizes the number of independent directions in which the space can be continuously deformed. A circle is a one-dimensional space because it can be continuously deformed into a line segment.

A homeomorphism is a continuous bijection between two topological spaces, meaning that it is a continuous function that is also one-to-one and onto. Homeomorphisms preserve the topological properties of the spaces they map between.

The fundamental group of a torus is the group of homotopy classes of loops based at a fixed point on the torus. It is isomorphic to $\mathbb{Z} \times \mathbb{Z}$, which means that it is generated by two independent loops that can be continuously deformed around the torus without intersecting each other.

A compact space is a topological space in which every open cover has a finite subcover. The closed interval $[0, 1]$ is compact because any open cover of $[0, 1]$ can be reduced to a finite number of open intervals that still cover $[0, 1]$.

The Euler characteristic of a space is a topological invariant that is defined as the alternating sum of the number of vertices, edges, and faces in a triangulation of the space. The Euler characteristic of a sphere is 2 because it can be triangulated into two triangles.

A connected space is a topological space in which any two points can be connected by a continuous path. The closed interval $[0, 1]$ is connected because any two points in $[0, 1]$ can be connected by a straight line segment.

The Hausdorff dimension of a set is a measure of its fractal dimension. The Cantor set is a fractal set that is constructed by repeatedly removing the middle third of each interval in a sequence of nested intervals. The Hausdorff dimension of the Cantor set is log(3)/log(2), which is approximately 0.6309.

A topological manifold is a space that is locally homeomorphic to Euclidean space. A sphere, torus, Klein bottle, and Möbius strip are all topological manifolds because they can be locally flattened out into Euclidean space.

The Poincaré duality theorem is a fundamental result in algebraic topology that relates the homology and cohomology groups of a topological space. It states that, under certain conditions, the homology groups of a space are isomorphic to the cohomology groups of the same space.

Homology groups are topological invariants that are used to study the shape and structure of topological spaces. The singular homology groups of a space are defined using singular simplices, which are continuous maps from the standard simplex to the space.

The homology dimension of a space is the largest integer n such that the nth homology group of the space is nonzero. The homology dimension of a circle is 1 because the first homology group of a circle is nonzero, while all higher homology groups are zero.

Cohomology groups are topological invariants that are used to study the shape and structure of topological spaces. The singular cohomology groups of a space are defined using singular cochains, which are continuous functions from the space to a coefficient group.

The de Rham cohomology theorem is a fundamental result in differential geometry that relates the de Rham cohomology groups of a smooth manifold to its singular cohomology groups. It states that, under certain conditions, the de Rham cohomology groups of a smooth manifold are isomorphic to the singular cohomology groups of the same manifold.