Relatively compact subset

A relatively compact subset (or precompact subset ) is a term from the mathematical sub-area of topology . It is a weakening of the topological concept of compact space .

definition

A subset of a topological space is relatively compact if their topological degree in compact. itself does not have to be compact. If, however, is already a closed subset of , is , then is a compact subset of . ${\ displaystyle A}$ ${\ displaystyle X}$ ${\ displaystyle {\ overline {A}}}$ ${\ displaystyle X}$ ${\ displaystyle A}$ ${\ displaystyle A}$ ${\ displaystyle X}$ ${\ displaystyle A = {\ overline {A}}}$ ${\ displaystyle A}$ ${\ displaystyle X}$

Some authors describe a relatively compact means . ${\ displaystyle A \ subset X}$ ${\ displaystyle A \ subset \ subset X}$

Other characterizations

Let it be a (in applications often: open) subset. A subset is relatively compact in if and only if is bounded and the closure of in does not meet the boundary of .

{\ displaystyle U \ subseteq \ mathbb {R} ^ {n}}

{\ displaystyle A \ subseteq U}

{\ displaystyle U}

{\ displaystyle A}

{\ displaystyle A}

{\ displaystyle \ mathbb {R} ^ {n}}

{\ displaystyle U}

In general, let it be a subset of a Hausdorff space and a subset of ; further is the conclusion of in . Then in is relatively compact if and only if compact and is contained in. ${\ displaystyle U}$ ${\ displaystyle X}$ ${\ displaystyle A}$ ${\ displaystyle U}$ ${\ displaystyle {\ bar {A}}}$ ${\ displaystyle A}$ ${\ displaystyle X}$ ${\ displaystyle A}$ ${\ displaystyle U}$ ${\ displaystyle {\ bar {A}}}$ ${\ displaystyle U}$

A subset of a metric space is relatively compact if and only if every sequence has in a convergent subsequence. ${\ displaystyle A}$ ${\ displaystyle X}$ ${\ displaystyle A}$ ${\ displaystyle X}$

An example

Let us use a set of real numbers as an example (with the usual Euclidean topology). Such a set of real numbers is compact if every infinite sequence of numbers from this set contains an infinite subsequence that comes “arbitrarily close” to another number, whereby this additional number must also belong to this set.

The amount of all real numbers between and (but without the boundary points and ) is not compact, because the infinite sequence , , , ... is indeed the accumulation point close arbitrarily, but does not belong more to (the same applies to all sub-sequences) . ${\ displaystyle A = (0.2)}$ ${\ displaystyle 0}$ ${\ displaystyle 2}$ ${\ displaystyle 0}$ ${\ displaystyle 2}$ ${\ displaystyle 1/1}$ ${\ displaystyle 1/2}$ ${\ displaystyle 1/3}$ ${\ displaystyle 1/4}$ ${\ displaystyle 0}$ ${\ displaystyle 0}$ ${\ displaystyle A}$

But what about the relative compactness of in if is the set of all real numbers? To zoom into a compact amount of accumulation points need and (the sequence , , , ... arbitrarily close to that) be added. In this way we get the closure of , that is the set of all real numbers from to (including these two boundary points). In fact, this degree is compact, so it is relatively compact in . ${\ displaystyle A}$ ${\ displaystyle U}$ ${\ displaystyle U}$ ${\ displaystyle A}$ ${\ displaystyle 0}$ ${\ displaystyle 2}$ ${\ displaystyle 1/1}$ ${\ displaystyle 3/2}$ ${\ displaystyle 5/3}$ ${\ displaystyle 7/4}$ ${\ displaystyle A}$ ${\ displaystyle [0,2]}$ ${\ displaystyle 0}$ ${\ displaystyle 2}$ ${\ displaystyle A}$ ${\ displaystyle U}$

While there are no edge points for ( ), the set of all positive real numbers has the edge point (which does not belong to ). Because the closure meets this edge point, the closure of in is equal to the set of all real numbers between (excluding) and (including). However, this set is not compact (because it again lacks the accumulation point ), so it is not relatively compact in . ${\ displaystyle X}$ ${\ displaystyle X = {\ mathbb {R}}}$ ${\ displaystyle X _ {+}}$ ${\ displaystyle 0}$ ${\ displaystyle X _ {+}}$ ${\ displaystyle [0,2]}$ ${\ displaystyle A}$ ${\ displaystyle X _ {+}}$ ${\ displaystyle (0.2]}$ ${\ displaystyle 0}$ ${\ displaystyle 2}$ ${\ displaystyle 0}$ ${\ displaystyle A}$ ${\ displaystyle X _ {+}}$