particular point topology

; Closed sets have empty interior

: Given a nonempty open set $A\; \backslash subseteq\; X$ every $x\; \backslash ne\; p$ is a limit point of A. So the closure of any open set other than $\backslash emptyset$ is $X$. No closed set other than $X$ contains p so the interior of every closed set other than $X$ is $\backslash emptyset$.

;Path and locally connected but not arc connected

For any x, y ∈ X, the function f: [0, 1] → X given by

:

p & t\in(0,1) \\

y & t=1

\end{cases}

is a path. 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, and 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 Sierpiński space then no such a and b exist and X is in fact ultraconnected.

; Compact only if finite. Lindelöf only if countable.

: If X is finite, it is compact; and if X is infinite, it is not compact, since the family of all open sets $\backslash \{p,x\backslash \}\backslash ;(x\backslash in\; X)$ forms an open cover with no finite subcover.

: For similar reasons, if X is countable, it is a Lindelöf space; and if X is uncountable, it is not Lindelöf.

; 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. 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 locally relatively compact.

: If $x\backslash in\; X$, then the set $\backslash \{x,p\backslash \}$ is a compact neighborhood of x. However the closure of this neighborhood is all of X, and hence if X is infinite, x does not have a closed compact neighborhood, and X is not locally relatively compact.

; Accumulation points of sets

: If $Y\backslash subseteq\; X$ does not contain p, Y has no accumulation point (because Y is closed in X and discrete in the subspace topology).

: If $Y\backslash subseteq\; X$ contains p, every point $x\backslash ne\; p$ is an accumulation point of Y, since $\backslash \{x,p\backslash \}$ (the smallest neighborhood of $x$) meets Y. Y has no ω-accumulation point. Note that p is never an accumulation point of any set, as it is isolated in X.

; Accumulation point as a set but not as a sequence

: Take a sequence $(a\_n)\_n$ of distinct elements that also contains p. The underlying set $\backslash \{a\_n\backslash \}$ has any $x\backslash ne\; p$ as an accumulation point. However the sequence itself has no accumulation point as a sequence, as the neighbourhood $\backslash \{y,p\backslash \}$ of any y cannot contain infinitely many of the distinct $a\_n$.

; T₀

:X is T₀ (since {x, p} is open for each x) but satisfies no higher separation axioms (because all non-empty open sets must contain p).

; Not regular

:Since every non-empty 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 non-empty open set contains p, no non-empty closed sets can be separated by neighbourhoods from each other, and thus X is not normal. Exception: the Sierpiński 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.

; Alexandrov-discrete

: The topology is an Alexandrov topology. The smallest neighbourhood of a point $x$ is $\backslash \{x,p\backslash \}.$

; Comparable (Homeomorphic topologies on the same set that are not comparable)

: Let $p,\; q\; \backslash in\; X$ with $p\; \backslash ne\; q$. Let $t\_p\; =\; \backslash \{S\; \backslash subseteq\; X\; \backslash mid\; p\backslash in\; S\backslash \}$ and $t\_q\; =\; \backslash \{S\; \backslash subseteq\; X\; \backslash mid\; q\backslash in\; S\backslash \}$. That is t_q is the particular point topology on X with q being the distinguished point. Then (X,t_p) and (X,t_q) are homeomorphic incomparable topologies on the same set.

; No nonempty dense-in-itself subset

: Let S be a nonempty subset of X. If S contains p, then p is isolated in S (since it is an isolated point of X). If S does not contain p, any x in S is isolated in S.

; 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, has the discrete topology.

• Alexandrov topology
• Excluded point topology
• Finite topological space
• List of topologies
• One-point compactification
• Overlapping interval topology

• {{Citation | last1=Steen | first1=Lynn Arthur | author1-link=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | author2-link=J. Arthur Seebach, Jr. | title=Counterexamples in Topology | orig-date=1978 | publisher=Springer-Verlag | location=Berlin, New York | edition=Dover reprint of 1978 | isbn=978-0-486-68735-3 |mr=507446 | year=1995}}

Category:Topological spaces