In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a topology induced from that of X called the subspace topology (or the relative topology, or the induced topology, or the trace topology).

Interior (topology)

1. In mathematics, specifically in topology, the interior of a subset S of a topological space X is the union of all subsets of S that are open in X.
2. If S is a subset of a Euclidean space, then x is an interior point of S if there exists an open ball centered at x which is completely contained in S.

interior point (plural interior points) (mathematics, topology) A point in a set that has a neighbourhood which is contained in .

The set Q of rational numbers has no interior or isolated points, and every real number is both a boundary and accumulation point of Q. Example 5.28.

Prove: The set of interior points of any set A, written int(A), is an open set. Let p∈ int(A), then by definition p must belong to some open interval Sp⊂A. Now since we know that the real line itself is open then Sp⊂R.

Say we want to show that the interior of a set A is open. If x∈Int(A), then there exists an open ball Br(x)⊆A. Since Br(x) is open, y∈Br(x) also has an open ball Bs(y)⊆Br(x)⊆A, so y∈Int(A).

To say that A has empty interior is to say then that A contains no open set of X other than the empty set. Equivalently, A has empty interior if every point of A is a limit point of the complement of A, that is, if the complement of A is dense in X.

(b) The interior of a set S is the set of all interior points of S (denoted by int(S)). (c) A set S is open iff every point in S is an interior point of S. (Since interior points are obviously elements of S, this is equivalent to int(S) = S.).

(i) intZ: By definition the interior of a set S are all the points x in it such that we can find an ε>0 such that B(x,ε)⊂S.

(e) Do E and E have the same interiors? Not necessarily. For instance (0, 1) ∪ (1, 2) ⊂ R is open, so equal to its interior, but its closure is [0, 2] with interior (0, 2) which contains the extra point 1. It is always the case that E◦ ⊂ (E)◦.

The set R∖Q is an example of a set which is uncountable, but it has empty interior. (No open interval is a subset of this set.)

The interior, or (open) kernel, of A is the set of all interior points of A: the union of all open sets of X which are subsets of A; a point x∈A is interior if there is a neighbourhood Nx contained in A and containing x. The interior may be denoted A∘, IntA or ⟨A⟩.

No, every interior point need not be a limit point. Hence there exists neighborhood V which contains in N hence by definition of an interior point, clearly ‘p’ is an interior point.

Edit: You can define a metric d(x,y)=1 if x≠y and d(x,y)=0 if x=y. So if (X,d) is a metric space, then you can show that every set is open and closed. Thus U⊂X is open so the boundary is in X−U and likewise X−U is open so the boundary is in U but (X−U)∩U=∅.

An open ball in the metric induced by ‖⋅‖ is a convex set.

An open ball in a metric space (X, ϱ) is an open set. Proof. If x ∈ Br(α) then ϱ(x, α) = r − ε where ε > 0.

an open ball in Rn is connected Suppose Br(xo) is not connected. Without loss of generality, let a∈Br(xo): a∈U. Since U is open, for some r1>0, Br1(xo)⊆U. Since (U∩Br(xo))∩(V∩Br(xo))=∅, a∉V.

One might be tempted to ask whether the closure of an open ball is equal to the corresponding closed ball . Unfortunately the answer is no in general.

Here the metric space is the circle with the top point removed. No. The Knaster-Kuratowski fan is a connected subspace of the plane that becomes totally disconnected when a certain point is removed, so open balls centred at the other points cannot be connected if they are small enough to exclude the explosion point.

The closed ball in Rn is an example of a manifold with boundary.

2.2 Examples (a) The Euclidean space Rn itself is a smooth manifold. One simply uses the identity map of Rn as a coordinate system.

However, for every point sufficiently small balls around it are convex.

Locally, the surface of the Earth looks like a 2-dimensional plane, so it is a 2-manifold.

The name manifold comes from Riemann’s original German term, Mannigfaltigkeit, which William Kingdon Clifford translated as “manifoldness”. As continuous examples, Riemann refers to not only colors and the locations of objects in space, but also the possible shapes of a spatial figure.

There are basically three possible shapes to the Universe; a flat Universe (Euclidean or zero curvature), a spherical or closed Universe (positive curvature) or a hyperbolic or open Universe (negative curvature).

Generally manifolds are taken to have a fixed dimension (the space must be locally homeomorphic to a fixed n-ball), and such a space is called an n-manifold; however, some authors admit manifolds where different points can have different dimensions. If a manifold has a fixed dimension, it is called a pure manifold.