# Particular point topology

In mathematics, the **particular point topology** (or **included point topology**) is a topology where sets are considered open if they are empty or contain a particular, arbitrarily chosen, point of the topological space. Formally, let *X* be any set and *p* ∈ *X*. The collection

*T*= {*S*⊆*X*:*p*∈*S*or*S*= ∅}

of subsets of *X* is then the particular point topology on *X*. There are a variety of cases which are individually named:

- If
*X*= {0,1} we call*X*the**Sierpiński space**. This case is somewhat special and is handled separately. - If
*X*is finite (with at least 3 points) we call the topology on*X*the**finite particular point topology**. - If
*X*is countably infinite we call the topology on*X*the**countable particular point topology**. - If
*X*is uncountable we call the topology on*X*the**uncountable particular point topology**.

A generalization of the particular point topology is the closed extension topology. In the case when *X* \ {*p*} has the discrete topology, the closed extension topology is the same as the particular point topology.

This topology is used to provide interesting examples and counterexamples.

## Properties

- Closed sets have empty interior
- Given an open set every is a limit point of A. So the closure of any open set other than is . No closed set other than contains p so the interior of every closed set other than is .

### Connectedness Properties

- Path and locally connected but not arc connected

*f*is a path for all*x*,*y*∈*X*. However since*p*is open, the preimage of*p*under a continuous injection from [0,1] would be an open single point of [0,1], which is a contradiction.

- Dispersion point, example of a set with
*p*is a**dispersion point**for*X*. That is*X\{p}*is totally disconnected.

- Hyperconnected but not ultraconnected
- Every non-empty open set contains
*p*hence*X*is hyperconnected. But if*a*and*b*are in*X*such that*p*,*a*, and*b*are three distinct points, then*{a}*and*{b}*are disjoint closed sets and thus*X*is not ultraconnected. Note that if*X*is the Sierpinski space then no such*a*and*b*exist and*X*is in fact ultraconnected.

### Compactness Properties

- Closure of compact not compact
- The set
*{p}*is compact. However its closure (the closure of a compact set) is the entire space*X*and if*X*is infinite this is not compact (since any set*{t,p}*is open). For similar reasons if*X*is uncountable then we have an example where the closure of a compact set is not a Lindelöf space.

- Pseudocompact but not weakly countably compact
- First there are no disjoint non-empty open sets (since all open sets contain 'p'). Hence every continuous function to the real line must be constant, and hence bounded, proving that
*X*is a pseudocompact space. Any set not containing*p*does not have a limit point thus if*X*if infinite it is not weakly countably compact.

- Locally compact but not strongly locally compact. Both possibilities regarding global compactness.
- If
*x ∈ X*then the set is a compact neighborhood of*x*. However the closure of this neighborhood is all of*X*and hence if*X*is infinite it is not strongly locally compact. - In terms of global compactness,
*X*finite if and only if*X*is compact. The first implication is immediate, the reverse implication follows from noting that is an open cover with no finite subcover.

### Limit related

- Accumulation point but not a ω-accumulation point
- If
*Y*is some subset containing*p*then any*x*different from*p*is an accumulation point of*Y*. However*x*is not an*ω-accumulation point*as {*x*,*p*} is one neighbourhood of*x*which does not contain infinitely many points from*Y*. Because this makes no use of properties of*Y*it leads to often cited counter examples.

- Accumulation point as a set but not as a sequence
- Take a sequence {
*a*_{i}} of distinct elements that also contains*p*. As in the example above, the underlying set has any*x*different from*p*as an accumulation point. However the sequence itself cannot possess accumulation point*y*for its neighbourhood {*y*,*p*} must contain infinite number of the distinct*a*_{i}.

### Separation related

- T
_{0} *X*is T_{0}(since {*x*,*p*} is open for each*x*) but satisfies no higher separation axioms (because all open sets must contain*p*).

- Not regular
- Since every nonempty open set contains
*p*, no closed set not containing*p*(such as*X*\{*p*}) can be separated by neighbourhoods from {*p*}, and thus*X*is not regular. Since complete regularity implies regularity,*X*is not completely regular.

- Not normal
- Since every nonempty open set contains
*p*, no nonempty closed sets can be separated by neighbourhoods from each other, and thus*X*is not normal. Exception: the Sierpinski topology is normal, and even completely normal, since it contains no nontrivial separated sets.

- Separability
*{p}*is dense and hence*X*is a separable space. However if*X*is uncountable then*X\{p}*is not separable. This is an example of a subspace of a separable space not being separable.

- Countability (first but not second)
- If
*X*is uncountable then*X*is first countable but not second countable.

- Comparable ( Homeomorphic topology on the same set that is not comparable)
- Let with . Let and . That is
*t*is the particular point topology on_{q}*X*with*q*being the distinguished point. Then*(X,t*and_{p})*(X,t*are homeomorphic incomparable topologies on the same set._{q})

- Density (no nonempty subsets dense in themselves)
- Let
*S*be a subset of*X*. If*S*contains*p*then S has no limit points (see limit point section). If*S*does not contain*p*then*p*is not a limit point of*S*. Hence*S*is not dense if*S*is nonempty.

- Not first category
- Any set containing
*p*is dense in*X*. Hence*X*is not a union of nowhere dense subsets.

- Subspaces
- Every subspace of a set given the particular point topology that doesn't contain the particular point, inherits the discrete topology.

## See also

## References

- Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978],
*Counterexamples in Topology*(Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 0507446

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