Definition:Topological Subspace

Topological Subspace
Let $$T = \left({A, \vartheta}\right)$$ be a topological space.

Let $$\varnothing \subset H \subseteq A$$ be a non-null subset of $$T$$.

Then the $$T_H = \left({H, \vartheta_H}\right)$$ is called a topological subspace of $$T$$.

The set $$\vartheta_H$$ is defined as $$\vartheta_H = \left\{{U \cap H: U \in \vartheta}\right\}$$, and is called the relative topology, the induced topology or the subspace topology on $$H$$.

The fact that $$T_H = \left({H, \vartheta_H}\right)$$ is a topological space is proved in Topological Subspace is a Topological Space.

Metric Subspace
Let $$\left({X, d}\right)$$ be a metric space.

Let $$Y \subseteq X$$.

Let $$d_Y: Y \times Y \to \reals$$ be the restriction $$d \restriction_{Y \times Y}$$ of $$d$$ to $$Y$$.

That is, let $$\forall x, y \in Y: d_Y \left({x, y}\right) = d \left({x, y}\right)$$.

The metric space axioms hold as well for $$d_Y$$ as they do for $$d$$.

Then $$d_Y$$ is a metric on $$Y$$ and is referred to as the metric induced on $$Y$$ by $$d$$.

The metric space $$\left({Y, d_Y}\right)$$ is called a metric subspace of $$\left({X, d}\right)$$.

Vector Subspace
Let $$K$$ be a division ring.

Let $$\left({S, +: \circ}\right)_K$$ be a $K$-algebraic structure with one operation.

Let $$T$$ be a closed subset of $$S$$.

Let $$\left({T, +_T: \circ_T}\right)_K$$ be an $K$-vector space where $$+_T$$ is the restriction of $$+$$ to $$T \times T$$ and $$\circ_T$$ is the restriction of $$\circ$$ to $$K \times T$$.

Then $$\left({T, +_T: \circ_T}\right)_K$$ is a (vector) subspace of $$\left({S, +: \circ}\right)_K$$.

Compare submodule.

Proper Subspace
If $$T$$ is a proper subset of $$S$$, then $$\left({T, +_T: \circ_T}\right)_K$$ is a proper (vector) subspace of $$\left({S, +: \circ}\right)_K$$.