Separation

• The points 0 and 1 are topologically distinguishable in S since ${\displaystyle \{1\}}$ is an open set which contains only one of these points. Therefore, S is a Kolmogorov (T₀) space.
• However, S is not T₁ since the point 1 is not closed. It follows that S is not Hausdorff, or T_n for any ${\displaystyle n\geq 1.}$
• S is not regular (or completely regular) since the point 1 and the disjoint closed set ${\displaystyle \{0\}}$ cannot be separated by neighborhoods. (Also regularity in the presence of T₀ would imply Hausdorff.)
• S is vacuously normal and completely normal since there are no nonempty separated sets.
• S is not perfectly normal since the disjoint closed sets ${\displaystyle \varnothing }$ and ${\displaystyle \{0\}}$ cannot be precisely separated by a function. Indeed, ${\displaystyle \{0\}}$ cannot be the zero set of any continuous function ${\displaystyle S\to \mathbb {R} }$ since every such function is constant.

Connectedness

• The Sierpiński space S is both hyperconnected (since every nonempty open set contains 1) and ultraconnected (since every nonempty closed set contains 0).
• It follows that S is both connected and path connected.
• A path from 0 to 1 in S is given by the function: ${\displaystyle f(0)=0}$ and ${\displaystyle f(t)=1}$ for ${\displaystyle t>0.}$ The function ${\displaystyle f:I\to S}$ is continuous since ${\displaystyle f^{-1}(1)=(0,1]}$ which is open in I.
• Like all finite topological spaces, S is locally path connected.
• The Sierpiński space is contractible, so the fundamental group of S is trivial (as are all the higher homotopy groups).

Compactness

• Like all finite topological spaces, the Sierpiński space is both compact and second-countable.
• The compact subset ${\displaystyle \{1\}}$ of S is not closed showing that compact subsets of T₀ spaces need not be closed.
• Every open cover of S must contain S itself since S is the only open neighborhood of 0. Therefore, every open cover of S has an open subcover consisting of a single set: ${\displaystyle \{S\}.}$
• It follows that S is fully normal.^[4]

Convergence

• Every sequence in S converges to the point 0. This is because the only neighborhood of 0 is S itself.
• A sequence in S converges to 1 if and only if the sequence contains only finitely many terms equal to 0 (i.e. the sequence is eventually just 1's).
• The point 1 is a cluster point of a sequence in S if and only if the sequence contains infinitely many 1's.
• Examples:
  • 1 is not a cluster point of ${\displaystyle (0,0,0,0,\ldots ).}$
  • 1 is a cluster point (but not a limit) of ${\displaystyle (0,1,0,1,0,1,\ldots ).}$
  • The sequence ${\displaystyle (1,1,1,1,\ldots )}$ converges to both 0 and 1.

Metrizability

• The Sierpiński space S is not metrizable or even pseudometrizable since every pseudometric space is completely regular but the Sierpiński space is not even regular.
• S is generated by the hemimetric (or pseudo-quasimetric) ${\displaystyle d(0,1)=0}$ and ${\displaystyle d(1,0)=1.}$

Other properties

• There are only three continuous maps from S to itself: the identity map and the constant maps to 0 and 1.
• It follows that the homeomorphism group of S is trivial.

Categorical description

The above construction can be described nicely using the language of category theory. There is a contravariant functor ${\displaystyle T:\mathbf {Top} \to \mathbf {Set} }$ from the category of topological spaces to the category of sets which assigns each topological space ${\displaystyle X}$ its set of open sets ${\displaystyle T(X)}$ and each continuous function ${\displaystyle f:X\to Y}$ the preimage map

$${\displaystyle f^{-1}:{\mathcal {T}}(Y)\to {\mathcal {T}}(X).}$$

The statement then becomes: the functor ${\displaystyle T}$ is represented by ${\displaystyle (S,\{1\})}$ where ${\displaystyle S}$ is the Sierpiński space. That is, ${\displaystyle T}$ is naturally isomorphic to the Hom functor ${\displaystyle \operatorname {Hom} (-,S)}$ with the natural isomorphism determined by the universal element ${\displaystyle \{1\}\in T(S).}$ This is generalized by the notion of a presheaf.^[6]

The initial topology

Any topological space X has the initial topology induced by the family ${\displaystyle C(X,S)}$ of continuous functions to Sierpiński space. Indeed, in order to coarsen the topology on X one must remove open sets. But removing the open set U would render ${\displaystyle \chi _{U}}$ discontinuous. So X has the coarsest topology for which each function in ${\displaystyle C(X,S)}$ is continuous.

The family of functions ${\displaystyle C(X,S)}$ separates points in X if and only if X is a T₀ space. Two points ${\displaystyle x}$ and ${\displaystyle y}$ will be separated by the function ${\displaystyle \chi _{U}}$ if and only if the open set U contains precisely one of the two points. This is exactly what it means for ${\displaystyle x}$ and ${\displaystyle y}$ to be topologically distinguishable.

Therefore, if X is T₀, we can embed X as a subspace of a product of Sierpiński spaces, where there is one copy of S for each open set U in X. The embedding map

$${\displaystyle e:X\to \prod _{U\in {\mathcal {T}}(X)}S=S^{{\mathcal {T}}(X)}}$$

is given by

$${\displaystyle e(x)_{U}=\chi _{U}(x).\,}$$

Since subspaces and products of T₀ spaces are T₀, it follows that a topological space is T₀ if and only if it is homeomorphic to a subspace of a power of S.