Lack of homeomorphism between compact space and non-Hausdorff space

Show that a continuous bijection $f : X \to Y$ with $X$ compact and $Y$ Hausdorff is a homeomorphism. Give an example to show that such a continuous bijection is not necessarily a homeomorphism if $Y$ is not assumed to be Hausdorff.

I'm having some trouble with the counterexample.

Would an $X$ interval in $\mathbb R$ and a $Y = S^1$ work?

-

1
$S^1$ is Hausdorff! – Grumpy Parsnip Oct 14 '11 at 1:05

3 Answers

No, because $\mathbb{S}^1$ is Hausdorff (it is a subspace of the Hausdorff space $\mathbb{R}^2$).

Let $X=\{a,b\}$ with the discrete topology, i.e. the open sets of $X$ are $\varnothing$, $\{a\}$, $\{b\}$, and $\{a,b\}$, and let $Y=\{c,d\}$ with the trivial topology, i.e. the open sets of $Y$ are $\varnothing$ and $\{c,d\}$.

$X$ is compact (because it is finite), and $Y$ is not Hausdorff because the points $c$ and $d$ cannot be separated by disjoint open sets.

Let the map $f:X\to Y$ be defined by $f(a)=c$, $f(b)=d$. Then $f$ is a continuous bijection, but not a homeomorphism.

-

Here is a simple example. Let $X$ be an interval with the standard topology, and let $Y$ be the same set with the coarse topology (only two open sets).

-

To show that f is a homeo. is to show that it takes open sets into open sets. Let $G$ be open in $X$. Then $F = X - G$ is closed and therefore compact (closed subspace of a compact space.) Since f is continuous, $f(F)$ is compact in $Y$. Since $Y$ is $T_2$ we have that $f(F)$ is closed in $Y$.

Then $Y - f(F) = f(G)\$ is open in Y. therefore $f^{-1}$ is continuous

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 41, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9477531313896179, "perplexity_flag": "head"}