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A Borel measure on a topological space is a measure defined on the sigma-algebra generated by the open sets, equivalently on the Borel sets.

A Borel set in a topological space belongs to the smallest sigma-algebra containing every open set.

The Borel sigma-algebra of a topological space $X$ is the sigma-algebra generated by its open sets. Its members are exactly the Borel sets.

The boundary of a subset $A$ of a topological space consists of the points whose every neighborhood meets both $A$ and its complement. Equivalently, $\partial A=\overline{A}\setminus A^{\circ }$, where the bar denotes closure and the circle denotes interior.

A point of a topological space is closed when its singleton is a closed set. In a T1 space every point is closed. The closed points of an affine scheme $\mathrm{Spec}A$ are its maximal ideals; the closure of the point corresponding to $\mathfrak{p}$ is $V(\mathfrak{p})$.

A compact manifold is a smooth manifold whose underlying topological space is compact. Every smooth vector field on a compact manifold without boundary has a flow defined for all real time. On a manifold with boundary, tangency to the boundary or an appropriate inward-flow condition must also be imposed.

A metric is compatible with a given topology when its open balls generate exactly that topology. Compatible metrics on a compact topological space are uniformly equivalent: the identity maps between the resulting compact metric spaces are uniformly continuous. Consequently they define the same proximality and the same asymptotic-pair relation for a fixed continuous map.

The product of connected spaces is connected, so the product of the two connected components lies in the component of $(x,y)$. Conversely, continuous projections carry that component to connected sets in the factors, which must lie in $C_x$ and $C_y$. These two inclusions prove equality in arbitrary topological spaces.

If a nonempty scheme $X$ has no nontrivial idempotent in $\Gamma (X,{\mathcal{O}}_X)$, then its underlying topological space is connected. Indeed, a decomposition into two nonempty clopen subsets would produce the global regular function equal to zero on one component and one on the other. In particular, $\Gamma (X,{\mathcal{O}}_X)$ being a field implies connectedness.

A curve in a topological space $X$ is a continuous function from an interval into $X$.

A family of sets has the finite intersection property when every finite subfamily has nonempty intersection. A topological space is compact exactly when every family of closed subsets with the finite intersection property has nonempty total intersection.

The Hawaiian earring is the union of a countable family of circles tangent at one point, with their radii tending to zero, endowed with its planar subspace topology. Equivalently it is the one-point compactification of a countable disjoint union of real lines. Every neighbourhood of the common point contains all but finitely many whole circles. This compact topological space is not locally contractible there and has different singular homology from the infinite CW complex wedge of circles.

The interior of a subset $A$ of a topological space is the union of all open sets contained in $A$, equivalently the largest open subset of $A$.

An irreducible scheme is one whose underlying topological space is a nonempty irreducible topological space. It can have nilpotent elements in its structure sheaf; irreducibility concerns only the topology. A reduced scheme is an integral scheme precisely when it is irreducible.

An algebraic variety whose underlying topological space with its Zariski topology is an irreducible topological space: it is nonempty and cannot be a union of two proper closed subsets. For a reduced affine variety, this is equivalent to its coordinate ring being an integral domain.

A topological space has local connectedness when every point has a neighborhood basis of connected open sets. In particular, a planar domain has local connectedness, but its domain boundary need not be. Local connectedness of the boundary is the condition governing a continuous extension of a conformal bijection to the parametrizing circle.

A family $(A_i)$ of subsets of a topological space is locally finite if every point has a neighbourhood meeting only finitely many of its members. On a compact space such a family has only finitely many nonempty members. For singleton subsets of a Riemann surface, this is the condition permitting prescribed poles of a meromorphic function without accumulation inside the surface.