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Take $\mathbb{R}\times \{0,1\}$ with its Euclidean topology and identify $(t,0)$ with $(t,1)$ for every $t\ne 0$, retaining two distinct origins. This quotient space is locally homeomorphic to the real line but is not Hausdorff. Any open neighbourhoods of its origins have preimages containing $(-\epsilon ,\epsilon )\times \{0\}$ and $(-\delta ,\delta )\times \{1\}$, so a nonzero $|t|<\min (\epsilon ,\delta )$ gives a shared image point. All equivalence classes in the original metric space are finite and closed. This shows that closed equivalence classes do not suffice for a Hausdorff quotient space.

Let $U$ be open in the Euclidean topology on the real line, and let $x\in U$. There is an $\epsilon >0$ with $(x-\epsilon ,x+\epsilon )\subseteq U$. The basic open set $[x,x+\epsilon )$ contains $x$ and is contained in $U$. Taking the union of these basic neighbourhoods over $x\in U$ shows that $U\in \mathcal{T}$. ThusIn fact this is a strictly finer topology: $[0,1)$ is open in $\mathcal{T}$ but not in the Euclidean topology, since no usual open ball about $0$ is contained in it.

Use the Heaviside step function with the value at the jump chosen as follows:For any open set in the target Sorgenfrey line, its preimage is one of $\varnothing$, $\mathbb{R}$, $(-\infty ,0)$, and $[0,\infty )$. All four are open in the source Sorgenfrey line, so $f$ is continuous. In the Euclidean topology, the preimage of $(1/2,3/2)$ is $[0,\infty )$, which is not open. Hence the same function is not continuous as a usual real-to-real map.

An abstract simplicial complex consists of a set $V$ of vertices of a simplicial complex and a family $K$ of finite nonempty subsets of $V$, called abstract simplexes, such that every singleton $\{v\}$ belongs to $K$ and every nonempty subset of a member of $K$ also belongs to $K$. One may include the empty set as an additional face by convention. A member with $k+1$ vertices is a $k$-simplex; its nonempty subsets are its faces of a simplex.

The geometric realization has the following explicit description using a barycentric coordinate system:Here the support is $\mathrm{supp}(t)=\{v:t(v)>0\}$. Each $\rho \in K$ gives a standard geometric simplexwith its usual Euclidean topology. Common faces of a simplex are identified using the same barycentric coordinate system.

Precisely, the topology on $|K|$ is defined by declaring $A\subseteq |K|$ closed if and only if $A\cap |\rho |$ is closed in $|\rho |$ for every $\rho \in K$. Equivalently, use the quotient topology on the disjoint union of all the standard geometric simplexes, identifying identical points of common faces of a simplex. For a finite simplicial complex, this agrees with the subspace topology in ${\mathbb{R}}^V$. For an infinite simplicial complex, the stated rule specifies the realization topology without presuming a finite-dimensional ambient space.