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.

Let $(X,\; \backslash mathcal\{T\})$ be a topological space and $P$ a set disjoint from $X$. The open extension topology of $\backslash mathcal\{T\}$ plus $P$ is $$\backslash mathcal\{T\}^*\; =\; \backslash mathcal\{T\}\; \backslash cup\; \backslash \{X\; \backslash cup\; A\; :\; A\; \backslash subset\; P\backslash \}.$$Let $X^*\; =\; X\; \backslash cup\; P$. Then $\backslash mathcal\{T\}^*$ is a topology in $X^*$. The subspace topology of $X$ is the original topology of $X$, i.e. $\backslash mathcal\{T\}^*|X\; =\; \backslash mathcal\{T\}$, while the subspace topology of $P$ is the discrete topology, i.e. $\backslash mathcal\{T\}^*|P\; =\; \backslash mathcal\{P\}(P)$.

The closed sets in $X^*$ are $\backslash \{B\; \backslash cup\; P\; :\; X\; \backslash subset\; B\; \backslash land\; X\; \backslash setminus\; B\; \backslash in\; \backslash mathcal\{T\}\backslash \}$. 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:{{sfn|Steen|Seebach|1995|p=48}}

• $X$ 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 – {p} plus p.

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 – {p} plus p.

{{Reflist}}

{{refbegin}}

• {{Citation| title = Counterexamples in Topology | edition = Dover reprint of 1978

| last1 = Steen | first1 = Lynn Arthur

| last2 = Seebach | first2 = J. Arthur Jr

| author1-link = Lynn Steen

| author2-link = J. Arthur Seebach Jr.

| year = 1995 | orig-year = First published 1978

| publisher = Springer-Verlag | location = Berlin, New York

| isbn = 978-0-486-68735-3 | mr = 507446

}}

{{refend}}

Category:Topological spaces

Category:Topology