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Let $G$ be an arbitrary group and we construct the classifying space $BG$ as quotient of $EG$ where the latter one is considered in this discussion to be constructed in natural way as $\Delta$-complex by method from Hatcher's AT book on page 89. That means all $n$-simplices of $EG$ are the ordered $(n + 1)$ tuples $[g_0, ..., g_n]$ of elements of $G$ glued together along faces $[g_0, ..., \hat{g_i},..., g_n]$ in canonical way. For concrete geometrical realization of such formally defined $n$-simplex $[g_0, ..., g_n]$ we identify $g_0,..., g_n$ with $(n+1)$ real points in $ \mathbb{R}^n$ which are affinely independent, which means $g_1-g_0,... g_n-g_0$ are linearly independent. Then we can write $[g_0, ..., g_n]$ as

$$ \{ \theta_0 g_0 + ... \theta_n g_n \ \vert \ \sum_{i=0}^n \theta_i=1 \text{ and } \theta_i \ge 0\} $$ and endow it naturally with Euklidian topology.
The group $G$ acts on $EG$ by left multiplication, an element $g \in G$ takes the simplex $[g_0, ..., g_n]$ linearly onto the simplex $[gg_0, ..., gg_n]$ and by definition $BG= EG/G$.

My goal is to understand locally the topology of $BG$ better. Is there a standard method known to construct open subsets in $BG$?
Note that since $BG$ inherits a $\Delta$-complex structure from $EG$, an subset $O \subset BG$ is open iff the intersection of $U$ with any simplex of $BG$ is open with respect to the induced standard Euklidian topology.

It seems naturally for me instead to work with complicated $BG$ to try to construct open subsets in $EG$, but assure that these are $G$-invariant with respect the described action.
Is there a most "natural, not too exotic" way (ways?) known to construct such open $G$-invariant subsets $O \in EG$. Moreover can these be arranged to form a cover of $EG$ and therefore that their images under $EG \to BG$ to form a cover of $BG$.

Contain such constructions also some geometric intuition or are these done pure axiomatically?

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    $\begingroup$ Do you have in mind $G$ to be a Lie group or a discrete one? $\endgroup$
    – Tommaso Scognamiglio
    Aug 7, 2021 at 15:47
  • $\begingroup$ One important thing to know is that there are several non-homeomorphic constructions of $BG$. Indeed $BG$ is usually only well-defined as a (pointed) homotopy type. This doesn't mean that the question you're asking cannot be interesting, but it is in some sense "strange", since it is about a specific model. $\endgroup$
    – Denis Nardin
    Aug 7, 2021 at 18:42
  • $\begingroup$ @TommasoScognamiglio: an abstract one, so with a discrete topology $\endgroup$
    – user267839
    Aug 7, 2021 at 20:58
  • $\begingroup$ @DenisNardin: I tried to choose a model that seems to carry the most elementary geometric structure. And simpler then that one of a $\Delta$-complex I don't know. But yes you are right that from the homotopy theoretical viewpoint such approach has not the potential to provide deep insights. My motivation is just to get some geometric intuition of $BG$ by considering a specific model which seems to me to be most approachable from naive-geometric viewpoint. $\endgroup$
    – user267839
    Aug 7, 2021 at 21:08

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