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 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 X 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 {ai} 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 ai.
Separation related
- T0
- X is T0 (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 tq is the particular point topology on X with q being the distinguished point. Then (X,tp) and (X,tq) are homeomorphic incomparable topologies on the same set.
- 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
- Sierpiński space
- Excluded point topology
- Alexandrov topology
- Finite topological space
- One-point compactification
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 507446
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