What is the name of this topology on the set of branches in a tree?

Question

In combinatorial set theory/set-theoretic topology, a tree is a partially ordered set $(T, <)$ s.t. for all $x$, the downset $\{ y \in T: y < x\}$ is wellordered (the order type of this set is called the height of $x$). A branch of $T$ is a maximal chain in $T$. I write $[T]$ for the set of branches of $T$. I think of $[T]$ as a topological space whose clopen basis is $\{[[x]] : x \in T\}$, where $[[x]]$ is the set of branches containing $x$.

If the height of each node in $T$ is finite, the topology of $[T]$ coincides with the product topology. What is the topology of $[T]$ called if each node possibly has infinite height, and what is a good source on this topological space? Obvious names like tree topology or branch space have been taken for something else. Tree topology is a topology on $T$ itself. A branch space's underlying set is of the form $[T]$, but its topology depend on additional data about $T$.

Answer 1

The space of branches with this topology has been called simply the branch space of the tree. This is the term used, for instance, by Luther Bush Fuller in Trees and proto-metrizable spaces [PDF] and by Yuan-Qing Qiao and Frank Tall in Perfectly normal non-metrizable non-Archimedean spaces are generalized Souslin lines [PDF]. Gary Gruenhage used the same term in his chapter Generalized Metric Spaces in the Handbook of Set-Theoretic Topology, though he enclosed it in scare quotes. However, Stevo Todorčević uses the same term for the space of branches with a somewhat different topology in his chapter Trees and linearly ordered sets in the same handbook, and Hal Bennett, Dave Lutzer, and Mary Ellen Rudin also use it for this topology in Lines, Trees, and Branch Spaces [PDF].

The term bounded topology that Pedro Sánchez Terraf found also appears in Souslin quasi-orders and definable cardinality [PDF], by Alessandro Andretta and Luca Motto Ros, Questions on Generalized Baire Spaces [PDF], by Yurii Khomskii, Giorgio Laguzzi, Benedikt Löwe, and Ilya Sharankou, and The Hurewicz dichotomy for generalized Baire spaces [PDF], by Philipp Lücke, Luca Motto Ros, and Philipp Schlicht, among other places.

Answer 2

Any tree can be embedded in some full $\kappa$-ary tree $\kappa^{<\kappa}$, and its set of branches will be a subset of the generalized Baire space $\kappa^\kappa$. The usual topology on $\kappa^\kappa$, called bounded topology in this paper, has a base of the form $$ [s] := \{f \in \kappa^\kappa : f \restriction |s| = s\} $$ where $s\in\kappa^{<\kappa}$. These correspond to your sets $[[s]]$ via the natural embedding $T\hookrightarrow \kappa^{<\kappa}$.

It seems that that this term is also used in algebraic topology but I doubt the two concepts are related.