topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
analysis (differential/integral calculus, functional analysis, topology)
metric space, normed vector space
open ball, open subset, neighbourhood
convergence, limit of a sequence
compactness, sequential compactness
continuous metric space valued function on compact metric space is uniformly continuous
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(continuous metric space valued function on compact metric space is uniformly continuous)
Let $K$ and $Y$ be two metric spaces regarded as topological spaces via their metric topology, such that $K$ compact.
Then every continuous function of the form
In particular since subspaces of metric spaces canonically are metric spaces, it follows that if $X$ is a metric space
Assume on the contrary that it were not. This would mean that for all $\epsilon \in (0,\infty)$ there would for all $n \in \mathbb{N}$ be points $x_n,y_n \in K$ with $d_K(x_n,y_n) \lt 1/(n+1)$ but $d_Y(f(x_n),f(y_n)) \gt \epsilon$.
Since in compact metric spaces every sequence has a converging subsequence (here), it follows that there is a converging subsequences $(x_{n_k})_{k \in \mathbb{N}}$. Since $d_K(x_{n_k} y_{n_k}) \lt 1/(n+1)$ it follows that also $(y_{n_k})_{k \in \mathbb{N}}$ is a converging subsequence which converges to the same point:
But since continuous functions preserve limits of sequences, it would follow that
This however would contradict the assumption that $d_Y(f(x_k), f(y_k)) \gt \epsilon$. Hence we have a proof by contradiction.
maps from compact spaces to Hausdorff spaces are closed and proper
sequentially compact metric spaces are equivalently compact metric spaces
compact spaces equivalently have converging subnet of every net
countably compact metric spaces are equivalently compact metric spaces
Last revised on June 21, 2017 at 04:48:49. See the history of this page for a list of all contributions to it.