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Let $\tau$ be a topology on $X$ such that for every $a, b$ in $X$ there exists a bijective function $f$ that is continuous and has a continuous inverse such that $f(a) = b$.

Examples of such a topology:
the topology induced by the euclidean metric on $\mathbb R^n$
the discrete topology, the trivial topology

non examples: $\tau = \{\emptyset,a,\{b,c\},X\}$ where $X = \{a,b,c\}$

Is there a name for these type of topologies?

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    $\begingroup$ It seems that such spaces are called homogeneous. $\endgroup$ – Sangchul Lee Dec 28 '17 at 20:56
  • $\begingroup$ yes that appears to be what I want, thank you, feel free to write that as an answer to the question $\endgroup$ – mathew Dec 28 '17 at 21:00
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As Sangchul says in the comments, such spaces are called homogeneous. An equivalent way to phrase the definition is that the group of homeomorphisms $X \to X$ acts transitively on $X$. Important examples include

  • any connected manifold (this is not obvious)
  • any topological group $G$
  • any quotient $G/H$ of a topological group by a subgroup.

The term "homogeneous space" is also used for something more specific, namely a pair consisting of a space $X$ and a topological (sometimes Lie) group $G$ acting transitively on it.

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