open extension topology

In topology, a branch of mathematics, an extension topology is a topology placed on the disjoint union of a topological space and another set. There are various types of extension topology, described in the sections below.

Extension topology

Let ''X'' be a topological space and ''P'' a set disjoint from ''X''. Consider in ''X'' ∪ ''P'' the topology whose open sets are of the form ''A'' ∪ ''Q'', where ''A'' is an open set of ''X'' and ''Q'' is a subset of ''P''.

The closed sets of ''X'' ∪ ''P'' are of the form ''B'' ∪ ''Q'', where ''B'' is a closed set of ''X'' and ''Q'' is a subset of ''P''.

For these reasons this topology is called the extension topology of ''X'' plus ''P'', with which one extends to ''X'' ∪ ''P'' the open and the closed sets of ''X''. As subsets of ''X'' ∪ ''P'' the subspace topology of ''X'' is the original topology of ''X'', while the subspace topology of ''P'' is the discrete topology. As a topological space, ''X'' ∪ ''P'' is homeomorphic to the topological sum of ''X'' and ''P'', and ''X'' is a clopen subset of ''X'' ∪ ''P''.

If ''Y'' is a topological space and ''R'' is a subset of ''Y'', one might ask whether the extension topology of ''Y'' – ''R'' plus ''R'' is the same as the original topology of ''Y'', and the answer is in general no.

Note the similarity of this extension topology construction and the Alexandroff one-point compactification, in which case, having a topological space ''X'' which one wishes to compactify by adding a point ∞ in infinity, one considers the closed sets of ''X'' ∪  to be the sets of the form ''K'', where ''K'' is a closed compact set of ''X'', or ''B'' ∪ , where ''B'' is a closed set of ''X''.

Open extension topology

Let $(X, \mathcal)$ be a topological space and $P$ a set disjoint from $X$. The open extension topology of $\mathcal$ plus $P$ is $\mathcal^* = \mathcal \cup \.$Let $X^* = X \cup P$. Then $\mathcal^*$is a topology in $X^*$. The subspace topology of ''$X$'' is the original topology of ''$X$'', i.e. $\mathcal^*, X = \mathcal$, while the subspace topology of ''$P$'' is the discrete topology, i.e. $\mathcal^*, P = \mathcal(P)$.

The closed sets in $X^*$ are $\$. Note that ''$P$'' is closed in $X^*$ and ''$X$'' is open and dense in $X^*$.

If ''Y'' a topological space and ''R'' is a subset of ''Y'', one might ask whether the open extension topology of ''Y'' – ''R'' plus ''R'' is the same as the original topology of ''Y'', and the answer is in general no.

Note that the open extension topology of $X^*$ is smaller than the extension topology of $X^*$.

Assuming ''$X$'' and ''$P$'' are not empty to avoid trivialities, here are a few general properties of the open extension topology:
* ''$X$'' is dense in $X^*$.
* If ''$P$'' is finite, $X^*$ is compact. So $X^*$ is a compactification of ''$X$'' in that case.
* $X^*$ is connected.
* If ''$P$'' has a single point, $X^*$ is ultraconnected.

For a set ''Z'' and a point ''p'' in ''Z'', one obtains the excluded point topology construction by considering in ''Z'' the discrete topology and applying the open extension topology construction to ''Z'' – plus ''p''.

Closed extension topology

Let ''X'' be a topological space and ''P'' a set disjoint from ''X''. Consider in ''X'' ∪ ''P'' the topology whose closed sets are of the form ''X'' ∪ ''Q'', where ''Q'' is a subset of ''P'', or ''B'', where ''B'' is a closed set of ''X''.

For this reason this topology is called the closed extension topology of ''X'' plus ''P'', with which one extends to ''X'' ∪ ''P'' the closed sets of ''X''. As subsets of ''X'' ∪ ''P'' the subspace topology of ''X'' is the original topology of ''X'', while the subspace topology of ''P'' is the discrete topology.

The open sets of ''X'' ∪ ''P'' are of the form ''Q'', where ''Q'' is a subset of ''P'', or ''A'' ∪ ''P'', where ''A'' is an open set of ''X''. Note that ''P'' is open in ''X'' ∪ ''P'' and ''X'' is closed in ''X'' ∪ ''P''.

If ''Y'' is a topological space and ''R'' is a subset of ''Y'', one might ask whether the closed extension topology of ''Y'' – ''R'' plus ''R'' is the same as the original topology of ''Y'', and the answer is in general no.

Note that the closed extension topology of ''X'' ∪ ''P'' is smaller than the extension topology of ''X'' ∪ ''P''.

For a set ''Z'' and a point ''p'' in ''Z'', one obtains the particular point topology construction by considering in ''Z'' the discrete topology and applying the closed extension topology construction to ''Z'' – plus ''p''.

Notes

Works cited

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Topological spaces
Topology